Symmetry Classes Research Articles

Classical symmetries and the Quantum Approximate Optimization Algorithm

We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular, we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques toward predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.

Low-energy effective theory and symmetry classification of flux phases on the kagome lattice

Motivated by recent experiments on AV$_3$Sb$_5$ (A=K,Rb,Cs), the chiral flux phase has been proposed to explain time-reversal symmetry breaking. To fully understand the physics behind the chiral flux phase, we construct a low-energy effective theory based on the van-Hove points around the Fermi surface. The possible symmetry-breaking states and their classifications of the low-energy effective theory are completely studied, especially the flux phases on Kagome lattice. In addition, we discuss the relations between the low-energy symmetry breaking orders, the chiral flux and charge bond orders. We find all possible 183 flux phases on Kagome lattice within 2*2 unit cell by brute-force approach and classify them by point group symmetry. Among the 183 phases, we find 3 classes in 1*1 unit cell, 8 classes in 1*2 unit cell and 18 classes in 2*2 unit cell, respectively. These results provide a full picture of the time-reversal symmetry breaking in Kagome lattices.

Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials

We study entire minimizers of the Allen–Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal Euler–Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of W at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified.

Topology of critical chiral phases: Multiband insulators and superconductors

Recent works have proved the existence of symmetry-protected edge states in certain one-dimensional topological band insulators and superconductors at the gap-closing points which define quantum phase transitions between two topologically nontrivial phases. We show how this picture generalizes to multiband critical models belonging to any of the chiral symmetry classes AIII, BDI, or CII of noninteracting fermions in one dimension.

Superdense coding in the resource theory of asymmetry

We consider a task in which classical information is encoded into a quantum system by an operation restricted by symmetry. The maximum amount of classical information that can be encoded under this restriction, namely the symmetry-restricted classical information capacity, depends on the initial state of the system. Our focus is on whether the capacity of an asymmetric state can be strictly larger than that of any symmetric states, whereas the latter is a strictly positive constant. That is, we ask whether an analog of superdense coding is implementable in the resource theory of asymmetry. We prove that superdense coding is implementable if and only if the unitary representation of the symmetry is non-Abelian and reducible. Thereby we provide an information theoretical classification of symmetries of quantum systems.

Superconductors with anomalous Floquet higher-order topology

We develop a general theory for two-dimensional (2D) anomalous Floquet higher-order topological superconductors (AFHOTSC), which are dynamical Majorana-carrying phases of matter with no static counterpart. Despite the triviality of its bulk Floquet bands, an AFHOTSC generically features the simultaneous presence of corner-localized Majorana modes at both zero and $\pi/T$ quasi-energies, a phenomenon beyond the scope of any static topological band theory. We show that the key to AFHOTSC is its unavoidable singular behavior in the phase spectrum of the bulk time-evolution operator. By mapping such evolution-phase singularities to the stroboscopic boundary signatures, we classify all 2D AFHOTSCs that are protected by a rotation group symmetry in symmetry class D. We further extract a higher-order topological index for unambiguously predicting the presence of Floquet corner Majorana modes, which we confirm numerically. Our theory serves as a milestone towards a dynamical topological theory for Floquet superconducting systems.

Prediction of walking Asymmetry using Spatiotemporal Gait Parameters and Guidance for fall risk Prediction or Geriatric Care: Experimental Study for Indian Population

Gait analysis allows the quantitative assessment of gait to recognize its associated variation and disorders. The reliability of analysis gets augmented when being compared with standard documented normative dataset. The purpose of this study is to establish the spatiotemporal gait parameters for the normative dataset for Indian Population. Eighty healthy subjects aged between 20 and 70 years with no impairments affecting gait, recorded their Footfall on self-selected walking speed on GAITRite® Electronic walkway. Successive ten iteration of Barefoot and Comfortable (specific to person) shoe wear walk are Considered to generate one record. Mean, standard deviation, coefficient of correlation, 95% Confidence Interval, and 95% Prediction Interval are calculated using descriptive statistics. Healthy Gait is often characterized as symmetric to verify, bilateral spatiotemporal parameters are considered and mean, standard Deviation, Variance, and Min-Max ranges are obtained. Chosen Spatial and temporal characteristics are taken into consideration separately to demonstrate role in diverse test cases to get the result. Obtained singular and bilateral ranges are recorded for classification of Symmetry and Asymmetry and based on that algorithm is proposed for identifying Gait Asymmetry. Study of these ranges provides the guideline for geriatric care.

Equation of Rayleigh Noise Reduction Model for Medical Ultrasound Imaging: Symmetry Classification, Conservation Laws and Invariant Solutions

Medical ultrasound imaging provides images of the internal body for diagnosis. Speckle noise is a major problem degrading image quality. In this paper, a wide class of noise reduction models is considered from the group analysis point of view. Group classification is performed, and conservation laws and invariant solutions of the studied equation are presented.

Symmetry-breaking signatures of multiple Majorana zero modes in one-dimensional spin-triplet superconductors

We study the effects of various symmetry-breaking perturbations on the experimentally measurable signatures (such as conductance and Josephson response) of quasi-one-dimensional (quasi-1D) spin-triplet superconductors. In the first part of the paper, we numerically compute the zero and nonzero temperature conductances of the quasi-1D nanowires that host multiple Majorana zero modes. Following the discussion of the case of s-wave Rashba nanowires, we shift to the main focus, i.e., multichannel spin-triplet superconductors. Applying gate voltages (which changes the symmetry of the spin-orbit coupling) as well as magnetic fields to the nanowire, tunes the system between different symmetry classes by splitting the multiple Majorana zero modes. We study how the conductance tracks the topological invariants and the spectra in all these cases. In the second part of the paper, we study the effects of the symmetry-induced spectrum-breaking on the Andreev spectra of Josephson junctions. Similar to the case of the conductance studies, we find that the spectrum shows multiple zero-energy Andreev bound states in the highly symmetric case with mirror and chiral symmetries.

A supervised learning algorithm for interacting topological insulators based on local curvature

Topological order in solid state systems is often calculated from the integration of an appropriate curvature function over the entire Brillouin zone. At topological phase transitions where the single particle spectral gap closes, the curvature function diverges and changes sign at certain high symmetry points in the Brillouin zone. These generic properties suggest the introduction of a supervised machine learning scheme that uses only the curvature function at the high symmetry points as input data. { We apply this scheme to a variety of interacting topological insulators in different dimensions and symmetry classes. We demonstrate that an artificial neural network trained with the noninteracting data can accurately predict all topological phases in the interacting cases with very little numerical effort.} Intriguingly, the method uncovers a ubiquitous interaction-induced topological quantum multicriticality in the examples studied.

Higher-order topological superconductors based on weak topological insulators

High-order topological phases host robust boundary states at the boundary of the boundary, which can be interpreted from their boundary topology. In this work, considering the interplay between superconductors and magnetic fields to gap the surface states of three-dimensional weak topological insulators, we show that second-order topological superconductors (TSCs) featuring helical or chiral Majorana hinge modes and third-order TSC featuring Majorana corner modes can be realized. Remarkably, the higher-order TSCs in our models can be attributed to their certain boundaries, surfaces, or hinges, which naturally behave as first-order TSC in DIII or D symmetry class. Correspondingly, these higher-order TSCs can be characterized by the boundary first-order topological invariants, such as surface Chern numbers or surface $Z_2$ topological invariants for surface TSCs. Our models can effectively capture the topology of iron-based superconductors with desired inverted band structures and superconducting pairings.

Invariance of KMS states on graph C*-algebras under classical and quantum symmetry

AbstractWe study the invariance of KMS states on graph $C^{\ast }$-algebras coming from strongly connected and circulant graphs under the classical and quantum symmetry of the graphs. We show that the unique KMS state for strongly connected graphs is invariant under the quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism groups preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph $C^{\ast }$-algebra having more than one KMS state such that all of them are invariant under the action of classical automorphism group of the graph, but there is a unique KMS state which is invariant under the action of quantum automorphism group of the graph.

From operator statistics to wormholes

For a generic quantum many-body system, the quantum ergodic regime is defined as the limit in which the spectrum of the system resembles that of a random matrix theory (RMT) in the corresponding symmetry class. In this paper we analyse the time dependence of correlation functions of operators. We study them in the ergodic limit as well as their approach to the ergodic limit which is controlled by non-universal massive modes. An effective field theory (EFT) corresponding to the causal symmetry and its breaking describes the ergodic phase. We demonstrate that the resulting Goldstone-mode theory has a topological expansion, analogous to the one described in arXiv:2008.02271 with added operator sources, whose leading non-trivial topologies give rise to the universal ramp seen in correlation functions. The ergodic behaviour of operators in our EFT is seen to result from a combination of RMT-like spectral statistics and Haar averaging over wave-functions. Furthermore we analytically capture the plateau behaviour by taking into account the contribution of a second saddle point. Our main interest are quantum many-body systems with holographic duals and we explicitly establish the validity of the EFT description in the SYK-class of models, starting from their microscopic description. By studying the tower of massive modes above the Goldstone sector we get a detailed understanding of how the ergodic EFT phase is approached and derive the relevant Thouless time scales. We point out that the topological expansion can be reinterpreted in terms of contributions of bulk wormholes and baby-universes.

The Multipole Structure and Symmetry Classification of Even-Type Deviators Decomposed from the Material Tensor.

The number of distinct components of a high-order material/physical tensor might be remarkably reduced if it has certain symmetry types due to the crystal structure of materials. An nth-order tensor could be decomposed into a direct sum of deviators where the order is not higher than n, then the symmetry classification of even-type deviators is the basis of the symmetry problem for arbitrary even-order physical tensors. Clearly, an nth-order deviator can be expressed as the traceless symmetric part of tensor product of n unit vectors multiplied by a positive scalar from Maxwell’s multipole representation. The set of these unit vectors shows the multipole structure of the deviator. Based on two steps of exclusion, the symmetry classifications of all even-type deviators are obtained by analyzing the geometric symmetry of the unit vector sets, and the general results are provided. Moreover, corresponding to each symmetry type of the even-type deviators up to sixth-order, the specific multipole structure of the unit vector set is given. This could help to identify the symmetry types of an unknown physical tensor and possible back-calculation of the involved physical coefficients.

Localization dynamics from static and mobile impurities

We study the superfluid response and localization dynamics from static and mobile impurities. The superfluidity is formed in the rung-Mott phase of a bosonic ladder model producing spin-Meissner currents induced by a $\mathbb{U}(1)$ gauge field or a uniform magnetic field. Impurities are described through two-state systems which act as a two-peak random potential. An impurity sits either at the top or at the bottom of the ladder on each rung equally, producing a telegraph signal. The impurities-matter coupling gives rise to a classical Ising symmetry for static and mobile impurities associated to the inversion symmetry of the two legs of the ladder. From the decoupled rungs limit, we also identify a local $\mathbb{Z}_2$ gauge theory for mobile impurities. The properties of the system are studied from an effective quantum spin model including the possibility of four-body coupling in the limit of a strong interaction between bosons and impurities. Through analytical approaches and numerical exact diagonalization, we study the superfluid currents both in the weakly-coupled and strongly-coupled rungs limits for the bosons. In the weakly-coupled rungs situation, we find a smooth power-law localization whereas the strongly-coupled rungs limit produces a steep localization or insulating phase for various configurations of the two-peak random potential. In the strongly disordered situation, through entanglement and bipartite fluctuation measures, we also identify a many-body localization regime in time after a quench of the system when prepared in a N\' eel state.

Relationship between Unstable Point Symmetries and Higher-Order Approximate Symmetries of Differential Equations with a Small Parameter

The framework of Baikov–Gazizov–Ibragimov approximate symmetries has proven useful for many examples where a small perturbation of an ordinary or partial differential equation (ODE, PDE) destroys its local exact symmetry group. For the perturbed model, some of the local symmetries of the unperturbed equation may (or may not) re-appear as approximate symmetries. Approximate symmetries are useful as a tool for systematic construction of approximate solutions. For algebraic and first-order differential equations, to every point symmetry of the unperturbed equation, there corresponds an approximate point symmetry of the perturbed equation. For second and higher-order ODEs, this is not the case: a point symmetry of the original ODE may be unstable, that is, not have an analogue in the approximate point symmetry classification of the perturbed ODE. We show that such unstable point symmetries correspond to higher-order approximate symmetries of the perturbed ODE and can be systematically computed. Multiple examples of computations of exact and approximate point and local symmetries are presented, with two detailed examples that include a fourth-order nonlinear Boussinesq equation reduction. Examples of the use of higher-order approximate symmetries and approximate integrating factors to obtain approximate solutions of higher-order ODEs are provided.

Effects of hybridization and spin–orbit coupling to induce odd-frequency pairing in two-band superconductors

The effect of spin-independent hybridization potential and spin–orbit coupling on two-band superconductor with equal-time s-wave interband pairing order parameter is investigated theoretically. To study symmetry classes in two-band superconductors, the Gor’kov equations are solved analytically. By defining spin-singlet and spin-triplet s-wave order parameter due to two-band degree of freedom, the symmetry classes of Cooper pair are studied. For spin singlet case, it is shown that spin-independent hybridization generates Cooper pair that belongs to even-frequency spin singlet even-momentum even-band parity (ESEE) symmetry class for both intraband and interband pairing correlations. For spin-triplet order parameter, intraband pairing correlation generates odd-frequency spin triplet even-momentum even-band parity (OTEE) symmetry class, whereas interband pairing correlation generates even-frequency spin-triplet even-momentum odd-band parity (ETEO) class. For the spin singlet, spin–orbit coupling generates pairing correlation that belongs to odd-frequency spin-singlet odd-momentum even-band parity (OSOE) symmetry class and even-frequency spin-singlet even-momentum even-band parity (ESEE) for intraband and interband pairing correlation, respectively. In the spin-triplet case for intraband and interband correlation, spin–orbit coupling generates even-frequency spin-triplet odd-momentum even-band parity (ETOE) and even-frequency spin-triplet even-momentum odd-band parity (ETEO), respectively. Our results are in agreement with previous experimental and theoretical studies.

Generalized multifractality at spin quantum Hall transition

Generalized multifractality characterizes scaling of eigenstate observables at Anderson-localization critical points. We explore generalized multifractality in 2D systems, with the main focus on the spin quantum Hall (SQH) transition in superconductors of symmetry class C. Relations and differences with the conventional integer quantum Hall (IQH) transition are also studied. Using the field-theoretical formalism of non-linear sigma-model, we derive the pure-scaling operators representing generalizing multifractality and then “translate” them to the language of eigenstate observables. Performing numerical simulations on network models for SQH and IQH transitions, we confirm the analytical predictions for scaling observables and determine the corresponding exponents. Remarkably, the generalized-multifractality exponents at the SQH critical point strongly violate the generalized parabolicity of the spectrum, which implies violation of the local conformal invariance at this critical point.

Information theory based symmetry classifications of more or less 2D periodic patterns in Islamic building ornaments and Hans Hinterreiter's graphic art

Information theory based symmetry classifications of more or less 2D periodic patterns in Islamic building ornaments and Hans Hinterreiter's graphic art

Information theory-based plane symmetry classifications: revealing pseudo-symmetries in the presence of noise

Information theory-based plane symmetry classifications: revealing pseudo-symmetries in the presence of noise