Symmetry Operations Research Articles

From β-KB3O4F2 to α-KB3O4F2: Phase Transition-Directed Great Changes in Structure and Optical Anisotropy.

Herein, a new fluorooxoborate α-KB3O4F2 is synthesized successfully in a closed system. It crystallizes in the P21/n space group and features the rare one-dimensional (1D) zigzag [B6O8F4]∞ chains built by fundamental building blocks (FBBs) [B6O9F4]. To the best of our knowledge, KB3O4F2 is a unique example of inorganic anhydrous borate whose two polymorphs show 1D B-O/F chains constructed by different FBBs but the same symmetry operation. Different from β-KB3O4F2, the inconsistent arrangement of π-conjugated [B2O5] units in α-KB3O4F2 makes it exhibit a small birefringence (0.011@546 nm). Further, α-KB3O4F2 owns a deep-ultraviolet (DUV) cutoff edge (<200 nm), suggesting that it has a potential application as the zero-order waveplate material in the DUV region.

Noninvertible Symmetries Act Locally by Quantum Operations

of quantum field theories and many-body systems generalize the concept of symmetries by allowing noninvertible operations in addition to more ordinary invertible ones described by groups. The aim of this Letter is to point out that these noninvertible symmetries act on local operators by , i.e., completely positive maps between density matrices, which form a natural class of operations containing both unitary evolutions and measurements and play an important role in quantum information theory. This observation will be illustrated by the Kramers-Wannier duality of the one-dimensional quantum Ising chain, which is a prototypical example of noninvertible symmetry operations. Published by the American Physical Society 2024

Global shift symmetry on an ADM hypersurface: Toward emergent gravity

Generalized symmetries and their spontaneous breakdown serve as the fundamental concept to constrain the many-body entanglement structure, which allows us to characterize quantum phases of matter and emergent collective excitations. For example, emergent photons may be understood by spontaneous 1-form symmetry breaking, which results from a long-ranged entanglement structure between UV microscopic degrees of freedom. In this study, we show that emergent “gravity” may also arise in a similar fashion, where quotes have been used to emphasize that the symmetry-constrained gravitons show unconventional properties compared to usual gravitons. As the electric 1-form symmetry in Maxwell theory is realized as a global shift symmetry of the spatial component of the U(1) gauge field, generated by the electric field, we demonstrate that a constant shift of the Arnowitt-Deser-Misner (ADM) metric on the spatial hypersurface can be viewed as a global symmetry, generated by the ADM canonical momentum. Deriving a vector-type conserved charge from the variation of action, we construct a shift symmetry operator. Considering a Wick rotation, we demonstrate that a gravitational Wilson loop is charged under the action of this shift symmetry operator, which thus confirms the existence of a generalized global symmetry on the ADM hypersurface. Based on the Ward identity, we show that the spontaneous breaking of this global shift symmetry may give rise to a nonpropagating massless symmetric gauge field at the boundary of the hypersurface. Published by the American Physical Society 2024

HARNESSING GROUP THEORY FOR SIMPLIFIED ELECTRIC CIRCUIT ANALYSIS

Electric circuit analysis is fundamental to advancing modern technology, yet traditional methods often become complex and inefficient when applied to large-scale or intricate networks. Group theory, a branch of abstract algebra renowned for its capacity to exploit symmetry, has shown promise as a solution to this challenge. By applying group theoretical principles to circuit analysis, particularly through symmetry operations and transformation groups, engineers can simplify complex systems, reducing computational demands and revealing deeper insights into circuit behavior. Through systematic symmetry-based simplifications, group theory can decrease the number of equations required by up to 40% in certain configurations, enhancing both analytical efficiency and practical implementation. However, challenges include the mathematical complexity inherent to group theory, the limitations in analyzing highly irregular or nonlinear circuits, and the need for compatible computational tools. Recommendations include the integration of group theoretical approaches into standard circuit analysis software, focused training for electrical engineers, and further research to address the application of these methods in emerging technologies such as quantum and neuromorphic computing. Therefore, this review aims to elucidate the practical potential of group theory in transforming electric circuit analysis, presenting a streamlined approach that balances theoretical rigor with engineering feasibility.

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.

Revealing symmetries in quantum computing for many-body systems

Abstract We develop a method to deduce the symmetry properties of many-body Hamiltonians when they are prepared in Jordan-Wigner form in which they can act on multi-qubit states. &amp;#xD;Symmetries, such as point-group symmetries in molecules, are apparent in the standard second quantized form of the Hamiltonian.&amp;#xD;They are, however,&amp;#xD;masked when the Hamiltonian is translated into a Pauli matrix representation required for its operation on qubits.&amp;#xD;To reveal these symmetries we prove a general theorem that provides a straightforward method to calculate the transformation of Pauli tensor strings under symmetry operations.&amp;#xD;They are a subgroup of the Clifford group transformations and induce a corresponding group representation inside the symplectic matrices.&amp;#xD;We finally give a simplified derivation of an affine qubit encoding scheme which allows for the removal of qubits due to Boolean symmetries and thus reduces effort in quantum computations&amp;#xD;for many-body systems.

Plaquette models, cellular automata, and measurement-induced criticality

Abstract We present a class of two-dimensional randomized plaquette models, where the multi-spin interaction term, referred to as the plaquette term, is replaced by a single-site spin term with a probability of $1-p$. By varying $p$, we observe a ground state phase transition, or equivalently, a phase transition of the symmetry operator. We find that as we vary $p$, the symmetry operator changes from being extensive to being localized in space. These models can be equivalently understood as 1+1D randomized cellular automaton dynamics, allowing the 2D transition to be interpreted as a 1+1D dynamical absorbing phase transition. In this paper, our primary focus is on the plaquette term with three or five-body interactions, where we explore the universality classes of the transitions. Specifically, for the model with five-body interaction, we demonstrate that it belongs to the same universality class as the measurement-induced entanglement phase transition observed in 1+1D Clifford dynamics, as well as the boundary entanglement transition of the 2D cluster state induced by random bulk Pauli measurements. This work establishes a connection between transitions in classical spin models, cellular automata, and hybrid random circuits.&amp;#xD;&amp;#xD;

Effect of parity–time symmetry operation on entanglement distribution in two parallel Heisenberg spin chains

Abstract The effect of the parity–time ( PT ) symmetry operation on the entanglement transfer in two parallel Heisenberg spin chains is investigated. We give two schemes, i.e. pre- and post- PT symmetric operations, to demonstrate the performance of the PT symmetric operation on the entanglement dynamics of two qubits. In our investigation, we find that the effect of pre- PT symmetric operation schemes on the protection of entanglement of two qubits in the case of double spin chains is superior to that of the single spin chain case. Conversely, the effect of the post- PT symmetric operation scheme on the protection of entanglement of two qubits in the single spin chain case is better than that in the double spin chain case.

How symmetry helps to improve the estimation of the hyperfine splitting of torsional levels due to tunneling. The case of the HSSSH molecule

In this study, we analyze a series of molecules belonging to the C2V(M) molecular symmetry group which are characterized by several conformers. The use of molecular symmetry at each stage of calculating the energy of stationary torsional states is demonstrated. In particular, the importance is shown of preliminary symmetrization of physical characteristics of the molecules obtained by quantum chemical calculations. For the first time, symmetry-adapted basis functions for the diagonal kinetic coefficients are presented, which for the analyzed molecules do not satisfy all symmetry operations of the C2V(M) group. Using the HSSSH molecule as an example, it is shown how the full or partial accounting for molecular symmetry influences the calculated values of ultra-small tunneling splittings of the ground torsional states of the trans- and cis-conformers. It has also been established that the Hamiltonian matrix is characterized by symmetry which, when taken into account, makes it possible to halve the time of calculation of its elements.

Efficient Beam Manipulation With Phase Symmetry Operations on Transmitarrays for Flat-Top Beams

Efficient Beam Manipulation With Phase Symmetry Operations on Transmitarrays for Flat-Top Beams

General theory for discrete symmetry-breaking equilibrium states in quantum systems.

Spontaneous symmetrybreaking in phase transitions occurs when the system Hamiltonian is symmetric under a certain transformation, but the equilibrium states observed in nature are not. Here we build two noncommuting quantities from the order parameter of the transition and the symmetry operator that are constants of motion when such equilibrium states exist. Then, we derive a general equilibrium ensemble for the ordered phase and show that equilibrium states consisting of superpositions of different symmetry-breaking states, like positive and negative magnetized states, may exist. We propose an experimental realization of such equilibrium states with the state-of-the-art quantum technologies, and test it by means of numerical calculations. Finally, we show that a small symmetry-breaking perturbation in the Hamiltonian stabilizes the conservation of one of the two former quantities, implying that symmetry-breaking equilibrium states become stable even in small quantum systems.

Exploiting Non-Abelian Point-Group Symmetry to Estimate the Exact Ground-State Correlation Energy of Benzene in a Polarized Split-Valence Triple-Zeta Basis Set.

Local electronic-structure methods in quantum chemistry operate on the ability to compress electron correlations more efficiently in a basis of spatially localized molecular orbitals than in a parent set of canonical orbitals. However, many typical choices of localized orbitals tend to be related by selected, near-exact symmetry operations whenever a molecule belongs to a point group, a feature which remains largely unexploited in most local correlation methods. The present Letter demonstrates how to leverage a recent unitary protocol for enforcing symmetry properties among localized orbitals to yield a high-accuracy estimate of the exact ground-state correlation energy of benzene (D6h) in correlation-consistent polarized basis sets of both double- and triple-ζ quality. Through an initial application to many-body expanded full configuration interaction (MBE-FCI) theory, we show how molecular point-group symmetry can lead to computational savings that are inversely proportional to the order of the point group in a manner generally applicable to the acceleration of modern local correlation methods.

Layer stacking effect on structural, vibrational, and electronic properties of Janus-Ga2SeTe crystals

The van der Waals stacking of Janus Ga2SeTe quadruple-atomic layers leads to the formation of polytype bulk crystals, where the stacking order plays a critical yet elusive role in their fundamental properties. Here, we perform first-principles density functional theory calculations to investigate the stacking-dependent structural, electronic, and vibrational properties of four Ga2SeTe polytypes. Our results indicate that the γ polytype has two sub-phases of γ [ABC] and γ [ACB]. Lattice vibration studies manifest that symmetry operations depend on the stacking orders, in which the irreducible representations of optically active modes at Γ for the β, ε, γ, and δ phases are Γ = 3A1 + 4B1 + 4E2 + 3E1, Γ = 7E+7A1, Γ = 11A1 + 11E, and Γ = 8E2 + 7E1 + 8B1 + 7A1, respectively. Moreover, taking spin–orbit coupling into account, the estimated indirect bandgaps are 0.967, 0.926, 0.879, 0.925, and 0.950 eV for β, ε, γ [ABC], γ [ACB], and δ polytypes, respectively, in which the bandgap differences between indirect and direct transitions are within 10 meV. The Bader charge analysis indicates that the stacking order modulates the charge distribution among the compositional layers. These findings provide valuable insights for the development of optoelectronic devices based on Janus structures.

Piecewise modified Prandtl-Ishlinskii model for precise nano-positioning

Piezoelectric ceramic, an indispensable micro-nano actuator, has advantages of small size, high sensitivity and strong controllability, which is widely applied in various fields like micro-nano processing, medicine, chemistry and materials. However, its nonlinear and asymmetric hysteresis features directly affect its positioning accuracy. The classical and generalized Prandtl-Ishlinskii models with linear symmetry operators are commonly used to deal with these issues, but they still face difficulties in asymmetric systems. Therefore, a piecewise modified Prandtl-Ishlinskii (PMPI) model is proposed to compensate nonlinear and asymmetric hysteresis in this paper. The new nonlinear operator and piecewise nonlinear envelope function are introduced into the PMPI model. Independent right and left parts are adopted to describe hysteresis increasing and decreasing stages, while the threshold and weight vectors correspond to them. Weight vectors are extended and a new diagonal matrix is constructed for avoiding excessive parameters and computation. Simulations and experiments are shown that the PMPI model has good compensation performance and effectiveness, which has better fitting, higher accuracy, fewer parameters and computation, and can be well applied to the precise nano-positioning systems with less nonlinear and asymmetric errors.

Discrete Z4 Symmetry in Quantum Gravity

We consider the discrete Z4 symmetry i^, which takes place in the scenario of quantum gravity where the gravitational tetrads emerge as the order parameter—the vacuum expectation value of the bilinear combination of fermionic operators. Under this symmetry operation, i^, the emerging tetrads are multiplied by the imaginary unit, i^eμa=−ieμa. The existence of such symmetry and the spontaneous breaking of this symmetry are also supported by the consideration of the symmetry breaking scheme in the topological superfluid 3He-B. The order parameter in 3He-B is also the bilinear combination of the fermionic operators. This order parameter is the analog of the tetrad field, but it has complex values. The i^-symmetry operation changes the phase of the complex order parameter by π/2, which corresponds to the Z4 discrete symmetry in quantum gravity. We also considered the alternative scenario of the breaking of this Z4 symmetry, in which the i^-operation changes sign of the scalar curvature, i^R=−R, and thus the Einstein–Hilbert action violates the i^-symmetry. In the alternative scenario of symmetry breaking, the gravitational coupling K=1/16πG plays the role of the order parameter, which changes sign under i^-transformation.

(α+2)-Dimensional fractional evolution equation: group classification, symmetries, reduction and conservation laws

ABSTRACT A preliminary group classification based on symmetry operators is applied to study invariance properties of the time-fractional ( α + 2 ) -dimensional fractional evolution equation. The concepts of Riemann–Liouville and Caputo fractional derivatives are used in this study. The similarity variables obtained from symmetries and one-dimensional optimal systems of constructed Lie algebras are used in order to find the group-invariant solutions of the equation. Finally conservation laws of the equation are derived via a modified version of Noether’s theorem.

Symmetries of modified Dirac operators in supergravity flux backgrounds

Modifications of Dirac operators in supergravity flux backgrounds are considered. Modified spin curvature operators and squares of modified Dirac operators corresponding to Schrödinger-Lichnerowicz-like formulas are obtained for different types of flux modifications. Symmetry operators of modified massless and massive Dirac equations are found in terms of modified Killing-Yano and modified conformal Killing-Yano forms. Extra constraints for symmetry operators in terms of different types of fluxes and modified Killing-Yano forms are determined.

Invariant analysis of the multidimensional Martinez Alonso–Shabat equation

Abstract This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end.

Symmetries and Separation of Variables

In this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.

On the holographic dual of a topological symmetry operator

We study the holographic dual of a topological symmetry operator in the context of the AdS/CFT correspondence. Symmetry operators arise from topological field theories localized on a subspace of the boundary conformal field theory spacetime. We use bottom up considerations to construct the topological sector associated with their bulk counterparts. In particular, by exploiting the structure of entanglement wedge reconstruction we argue that the bulk counterpart has a nontopological world volume action, i.e., it describes a dynamical object. As a consequence, we find that there are no global p-form symmetries for p≥0 in asymptotically anti–de Sitter spacetimes, which includes the case of noninvertible symmetries. Provided one has a suitable notion of subregion-subregion duality, our argument for the absence of bulk global symmetries applies to more general spacetimes. These considerations also motivate us to consider for general QFTs (holographic or not) the notion of lower-form symmetries, namely, (−m)-form symmetries for m≥2. Published by the American Physical Society 2024