Quantum Symmetry Research Articles

Role of mixed permutation symmetry sectors in the thermodynamic limit of critical three-level Lipkin-Meshkov-Glick atom models.

We introduce the notion of mixed symmetry quantum phase transition (MSQPT) as singularities in the transformation of the lowest-energy state properties of a system of identical particles inside each permutation symmetry sector μ, when some Hamiltonian control parameters λ are varied. We use a three-level Lipkin-Meshkov-Glick model, with U(3) dynamical symmetry, to exemplify our construction. After reviewing the construction of U(3) unitary irreducible representations using Young tableaux and the Gelfand basis, we first study the case of a finite number N of three-level atoms, showing that some precursors (fidelity susceptibility, level population, etc.) of MSQPTs appear in all permutation symmetry sectors. Using coherent (quasiclassical) states of U(3) as variational states, we compute the lowest-energy density for each sector μ in the thermodynamic N→∞ limit. Extending the control parameter space by μ, the phase diagram exhibits four distinct quantum phases in the λ-μ plane that coexist at a quadruple point. The ground state of the whole system belongs to the fully symmetric sector μ=1 and shows a fourfold degeneracy, due to the spontaneous breakdown of the parity symmetry of the Hamiltonian. The restoration of this discrete symmetry leads to the formation of four-component Schrödinger cat states.

Berry curvature-induced emerging magnetic response in two-dimensional materials

The magnetic response in a two-dimensional material has received increasing attention in recent years. The magnetic effects and related quantum transport originate from Berry curvature, which is associated with crystal symmetry and many quantum effects including electrons’ orbital magnetism, spin-orbit coupling, and magnetoelectricity. The importance of studying the magnetic response in the two-dimensional material lies in two aspects. First, the magnetic response of two-dimensional material provides a platform to investigate the coupling between the above-mentioned intrinsic quantum effects and their couplings. Second, it possesses the potential applications in energy-efficient quantum and spintronic devices. Here, we review the experimental research progress made in recent years. In particular, we focus on the research progress of the valley Hall and magnetoelectric effect, quantum non-linear Hall effect, anomalous Hall, and quantum anomalous Hall effect in two-dimensional materials such as graphene, transition-metal chalcogenides, and twisted bilayer graphene. For each session, we first introduce these phenomena and their underlying physics by using crystal symmetries and band structures. Then, we summarize the experimental results and identify unsolved problems. At last, we provide an outlook in this emerging research direction.

Schwinger's picture of quantum mechanics: 2-groupoids and symmetries

<p style='text-indent:20px;'>Starting from the groupoid approach to Schwinger's picture of Quantum Mechanics, a proposal for the description of symmetries in this framework is advanced. It is shown that, given a groupoid <inline-formula><tex-math id="M1">\begin{document}$ G\rightrightarrows \Omega $\end{document}</tex-math></inline-formula> associated with a (quantum) system, there are two possible descriptions of its symmetries, one "microscopic", the other one "global". The microscopic point of view leads to the introduction of an additional layer over the grupoid <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>, giving rise to a suitable algebraic structure of 2-groupoid. On the other hand, taking advantage of the notion of group of bisections of a given groupoid, the global perspective allows to construct a group of symmetries out of a 2-groupoid. The latter notion allows to introduce an analog of the Wigner's theorem for quantum symmetries in the groupoid approach to Quantum Mechanics.</p>

Influence of temperature dependence and Li substitution on the electronic structure of delafossite CuFeO2 and CuFeS2 semiconductors

The electronic structure properties of the delafossite CuFeO2 and Chalcopyrite CuFeS2 semiconductors were studied by XAFS technique theoretically. Calculations were performed with the XAFS code FEFF 8.20 under increasing temperature conditions for 300 K, 350 K, 450 K, 500 K, and for 20% Li substitution at room temperature. High stability in the electronic structure of both materials addressed was observed as a result of the hybridized states built with the contribution of the overlapped outer shell electrons and levels of the neighboring atoms, i.e., Cu, Fe, O or S. The hybridized levels were determined as supporting the stability of the molecular bands via rich quantum symmetries. For the second step of the studies, 20% Li substitution in copper coordination was processed to probe the response of the materials to substitution mechanisms. In the CuFeO2 semiconductors, 0.5 eV shift was observed with the Li substitution at the Fe K-edge absorption spectra which caused a loose in the Li-O bonds as a result of the weaker electronegativity which increased the strength of the Fe-O bonds.

Continuous Symmetries and Approximate Quantum Error Correction

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here we study the compatibility of these two important principles. If a logical quantum system is encoded into $n$ physical subsystems, we say that the code is covariant with respect to a symmetry group $G$ if a $G$ transformation on the logical system can be realized by performing transformations on the individual subsystems. For a $G$-covariant code with $G$ a continuous group, we derive a lower bound on the error correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems $n$ or the dimension $d$ of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with $n$ or $d$ as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large $d$ (using random codes) or $n$ (using codes based on $W$-states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.

Partition quantum spaces

We propose a definition of partition quantum spaces. They are given by universal $C\*$-algebras whose relations come from partitions of sets. We ask for the maximal compact matrix quantum group acting on them. We show how those fit into the setting of easy quantum groups: Our approach yields spaces these groups are acting on. In a way, our partition quantum spaces arise as the first d columns of easy quantum groups. However, we define them as universal $C\*$-algebras rather than as $C\*$-subalgebras of easy quantum groups. We also investigate the minimal number $d$ needed to recover an easy quantum group as the quantum symmetry group of a partition quantum space. In the free unitary case, d takes the values one or two.

Multiplicities, Pictographs, and Volumes

The present contribution is the written counterpart of a talk given in Yerevan at the SQS'2019 International Workshop "Supersymmetries and Quantum Symmetries" (SQS'2019, 26 August - August 31, 2019). After a short presentation of various pictographs (O-blades, metric honeycombs) that one can use in order to calculate SU(n) multiplicities (Littlewood-Richardson coefficients, Kostka numbers), we briefly discuss the semi-classical limit of these multiplicities in relation with the Horn and Schur volume functions and with the so-called Rn-polynomials that enter the expression of volume functions. For n < 7 the decomposition of the Rn-polynomials on Lie group characters is already known, the case n=7 is obtained here.

Three-Pronged Junctions on SO(2N)/U(N) and Sp(N)/U(N)

We report on the results of our work [1], which discusses $$\mathcal{N} = 2$$ nonlinear sigma models on the quadrics of the Grassmann manifold and three-pronged junctions of the mass-deformed nonlinear sigma models on $$SO(8){\text{/}}U(4)$$ and $$Sp(3){\text{/}}U(3)$$ . This article is prepared for the Proceedings of International Workshop “Supersymmetries and Quantum Symmetries—SQS’2019”, which was held in Yerevan from 26 to 31 August, 2019. The talk was based on [1].

ΔK=0 M1 Excitation Strength of the Well-Deformed Nucleus Dy164 from K Mixing

The size of a ΔK=0 M1 excitation strength has been determined for the first time in a predominantly axially deformed even-even nucleus. It has been obtained from the observation of a rare K-mixing situation between two close-lying J^{π}=1^{+} states of the nucleus ^{164}Dy with components characterized by intrinsic projection quantum numbers K=0 and K=1. Nuclear resonance fluorescence induced by quasimonochromatic linearly polarized γ-ray beams provided evidence for K mixing of the 1^{+} states at 3159.1(3) and 3173.6(3)keV in excitation energy from their γ-decay branching ratios into the ground-state band. The ΔK=0 transition strength of B(M1;0_{1}^{+}→1_{K=0}^{+})=0.008(1)μ_{N}^{2} was inferred from a mixing analysis of their M1 transition rates into the ground-state band. It is in agreement with predictions from the quasiparticle phonon nuclear model. This determination represents first experimental information on the M1 excitation strength of a nuclear quantum state with a negative R-symmetry quantum number.

Quantum quench, large N, and symmetry restoration

We globally quench the theory of two dimensional massless fermions (many flavours) with quartic interactions by making the quartic coupling a smooth function of time. Working in a derivative expansion we show that the discrete ℤ2 symmetry in case of the Gross-Neveu model, and the U (1) symmetry in case of the Nambu-Jona-Lasinio2 model, are restored during the zero-temperature quench. For the Gross-Neveu model we show that this can be understood as an effective thermalization. The time of symmetry restoration shows scaling with the quench rate. We identify this with the Kibble-Zurek scaling in the problem. In a suitable double scaling limit, the symmetry restoration may be understood in terms of Liouville quantum mechanics.

Integrability Aspects of the Current Algebra Representation and the Factorized Quantum Nonlinear Schrödinger Type Dynamical Systems

In this work we study integrability aspects of the current algebra representation and the factorized quantum nonlinear Schrodinger type dynamical systems, initiated before by G. Goldin with collaborators, in suitably renormalized Fock type Hilbert spaces. There is developed the current algebra representation scheme of reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space. There is presented its application to constructing quantum factorized Hamiltonian systems and their symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially one-dimensional models as the Calogero–Moser–Sutherland and Nonlinear Schrodinger dynamical systems of spin-less Bose-particles.

Quantum automorphisms of folded cube graphs

We show that the quantum automorphism group of the Clebsch graph is $SO_5^{-1}$. This answers a question by Banica, Bichon and Collins from 2007. More general for odd $n$, the quantum automorphism group of the folded $n$-cube graph is $SO_n^{-1}$. Furthermore, we show that if the automorphism group of a graph contains a pair of disjoint automorphisms this graph has quantum symmetry.

Almost all trees have quantum symmetry

From the work of Erdős and Rényi from 1963, it is known that almost all graphs have no symmetry. In 2017, Lupini, Mančinska, and Roberson proved a quantum counterpart: Almost all graphs have no quantum symmetry. Here, the notion of quantum symmetry is phrased in terms of Banica’s definition of quantum automorphism groups of finite graphs from 2005, in the framework of Woronowicz’s compact quantum groups. Now, Erdős and Rényi also proved a complementary result in 1963: Almost all trees do have symmetry. The crucial point is the almost sure existence of a cherry in a tree. But even more is true: We almost surely have two cherries in a tree—and we derive that almost all trees have quantum symmetry. We give an explicit proof of this quantum counterpart of Erdős and Rényi’s result on trees.

The κ-Newtonian and κ-Carrollian algebras and their noncommutative spacetimes

We derive the non-relativistic c→∞ and ultra-relativistic c→0 limits of the κ-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the κ-(A)dS quantum algebra, and quantize the resulting contracted Poisson–Hopf algebras, thus giving rise to the κ-deformation of the Newtonian (Newton–Hooke and Galilei) and Carrollian (Para-Poincaré, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding κ-Newtonian and κ-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the κ-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter κ, the curvature parameter η and the speed of light parameter c.

Temperley–Lieb Quantum Channels

We study a class of quantum channels arising from the representation theory of compact quantum groups that we call Temperley-Lieb quantum channels. These channels simultaneously extend those introduced in [BC18], [AN14], and [LS14]. (Quantum) Symmetries in quantum information theory arise naturally from many points of view, providing an important source of new examples of quantum phenomena, and also serve as useful tools to simplify or solve important problems. This work provides new applications of quantum symmetries in quantum information theory. Among others, we study entropies and capacitites of Temperley-Lieb channels, their (anti-)degradability, PPT and entanglement breaking properties, as well as the behaviour of their tensor products with respect to entangled inpurs. Finally we compare the Tempereley-Lieb channels with the (modified) TRO-channels recently introduced in [GJL16].

π0 decay precision-tests the chiral anomaly.

More precise neutral pion lifetime measurements probe quantum symmetry breaking

Quantum Fourier analysis

Quantum Fourier analysis is a subject that combines an algebraic Fourier transform (pictorial in the case of subfactor theory) with analytic estimates. This provides interesting tools to investigate phenomena such as quantum symmetry. We establish bounds on the quantum Fourier transform F, as a map between suitably defined [Formula: see text] spaces, leading to an uncertainty principle for relative entropy. We cite several applications of quantum Fourier analysis in subfactor theory, in category theory, and in quantum information. We suggest a topological inequality, and we outline several open problems.

Non-existence of genuine (compact) quantum symmetries of compact, connected smooth manifolds

Suppose that a compact quantum group Q acts faithfully on a smooth, compact, connected manifold M, i.e. has a C⁎ (co)-action α on C(M), such that α(C∞(M))⊆C∞(M,Q) and the linear span of α(C∞(M))(1⊗Q) is dense in C∞(M,Q) with respect to the Fréchet topology. It was conjectured by the author quite a few years ago that Q must be commutative as a C⁎ algebra i.e. Q≅C(G) for some compact group G acting smoothly on M. The goal of this paper is to prove the truth of this conjecture. A remarkable aspect of the proof is the use of probabilistic techniques involving Brownian stopping time.

Ultrafast Control of Excitonic Rashba Fine Structure by Phonon Coherence in the Metal Halide Perovskite CH_{3}NH_{3}PbI_{3}.

We discover hidden Rashba fine structure in CH_{3}NH_{3}PbI_{3} and demonstrate its quantum control by vibrational coherence through symmetry-selective vibronic (electron-phonon) coupling. Above a critical threshold of a single-cycle terahertz pump field, a Raman phonon mode distinctly modulates the middle excitonic states with persistent coherence for more than ten times longer than the ones on two sides that predominately couple to infrared phonons. These vibronic quantum beats, together with first-principles modeling of phonon periodically modulated Rashba parameters, identify a threefold excitonic fine structure splitting, i.e., optically forbidden, degenerate dark states in between two bright ones with a narrow, ∼3 nm splitting. Harnessing of vibronic quantum coherence and symmetry inspires light-perovskite quantum control and sub-THz-cycle "Rashba engineering" of spin-split bands for ultimate multifunction device.

On the quantum symmetry of distance-transitive graphs

In this article, we study quantum automorphism groups of distance-transitive graphs. We show that the odd graphs, the Hamming graphs H(n,3), the Johnson graphs J(n,2) and the Kneser graphs K(n,2) do not have quantum symmetry. We also give a table with the quantum automorphism groups of all cubic distance-transitive graphs. Furthermore, with one graph missing, we can now decide whether or not a distance-regular graph of order ≤20 has quantum symmetry. Moreover, we prove that the Hoffman-Singleton graph has no quantum symmetry. On a final note, we present an example of a pair of graphs with the same intersection array (the Shrikhande graph and the 4×4 rook's graph), where one of them has quantum symmetry and the other one does not.