Residual Degeneracy Research Articles

The Virasoro-Shapiro amplitude in AdS5 \xd7 S5 and level splitting of 10d conformal symmetry

The genus zero contribution to the four-point correlator leftlangle {mathcal{O}}_{p_1}{mathcal{O}}_{p_2}{mathcal{O}}_{p_3}{mathcal{O}}_{p_4}rightrangle of half-BPS single-particle operators {mathcal{O}}_p in mathcal{N} = 4 super Yang-Mills, at strong coupling, computes the Virasoro-Shapiro amplitude of closed superstrings in AdS5× S5. Combining Mellin space techniques, the large p limit, and data about the spectrum of two-particle operators at tree level in supergravity, we design a bootstrap algorithm which heavily constrains its α′ expansion. We use crossing symmetry, polynomiality in the Mellin variables and the large p limit to stratify the Virasoro-Shapiro amplitude away from the ten-dimensional flat space limit. Then we analyse the spectrum of exchanged two-particle operators at fixed order in the α′ expansion. We impose that the ten-dimensional spin of the spectrum visible at that order is bounded above in the same way as in the flat space amplitude. This constraint determines the Virasoro-Shapiro amplitude in AdS5× S5 up to a small number of ambiguities at each order. We compute it explicitly for (α′)5,6,7,8,9. As the order of α′ grows, the ten dimensional spin grows, and the set of visible two-particle operators opens up. Operators illuminated for the first time receive a string correction to their anomalous dimensions which is uniquely determined and lifts the residual degeneracy of tree level supergravity, due to ten-dimensional conformal symmetry. We encode the lifting of the residual degeneracy in a characteristic polynomial. This object carries information about all orders in α′. It is analytic in the quantum numbers, symmetric under an AdS5 ↔ S5 exchange, and it enjoys intriguing properties, which we explain and detail in various cases.

Almost linear Haldane pseudopotentials and emergent conformal block wave functions in a Landau level

We consider a two dimensional (2D) model of particles interacting in a Landau level. We work in a finite disk geometry and take the particles to interact with a linearly decreasing two-body Haldane pseudo-potential. We show that the ground state subspace of this model is spanned by the wave-functions that can be written as polynomial conformal blocks (of an arbitrary conformal field theory) consistent with the filling fraction (scaling dimension). To remove degeneracies, we then add a quadratic perturbation to the Hamiltonian and show that; 1. Conformal blocks constructed using the Moore-Read construction (e.g. Laughlin, Pfaffian, and Read-Rezayi states) remain exact eigenstates of this model in the thermodynamic limit and 2. By tuning an externally imposed single-body $-L_z^2$ potential we can enforce Moore-Read conformal blocks to become exact ground states of this model in the thermodynamic limit. We cannot rule out the possibility of residual degeneracies in this limit. This model has no filling dependence and is comprised only from two-body long-range interactions and external single-body potentials. Our results provide insight into how conformal block wave-functions can emerge in a Landau level.

Double-trace spectrum of N=4 supersymmetric Yang-Mills theory at strong coupling

The spectrum of IIB supergravity on AdS${}_5 \times S^5$ contains a number of bound states described by long double-trace multiplets in $\mathcal{N}=4$ super Yang-Mills theory at large 't Hooft coupling. At large $N$ these states are degenerate and to obtain their anomalous dimensions as expansions in $\tfrac{1}{N^2}$ one has to solve a mixing problem. We conjecture a formula for the leading anomalous dimensions of all long double-trace operators which exhibits a large residual degeneracy whose structure we describe. Our formula can be related to conformal Casimir operators which arise in the structure of leading discontinuities of supergravity loop corrections to four-point correlators of half-BPS operators.

Models of the strongly lensed quasar DES J0408−5354

We present gravitational lens models of the multiply imaged quasar DES J0408-5354, recently discovered in the Dark Energy Survey (DES) footprint, with the aim of interpreting its remarkable quad-like configuration. We first model the DES single-epoch $grizY$ images as a superposition of a lens galaxy and four point-like objects, obtaining spectral energy distributions (SEDs) and relative positions for the objects. Three of the point sources (A,B,D) have SEDs compatible with the discovery quasar spectra, while the faintest point-like image (G2/C) shows significant reddening and a `grey' dimming of $\approx0.8$mag. In order to understand the lens configuration, we fit different models to the relative positions of A,B,D. Models with just a single deflector predict a fourth image at the location of G2/C but considerably brighter and bluer. The addition of a small satellite galaxy ($R_{\rm E}\approx0.2$") in the lens plane near the position of G2/C suppresses the flux of the fourth image and can explain both the reddening and grey dimming. All models predict a main deflector with Einstein radius between $1.7"$ and $2.0",$ velocity dispersion $267-280$km/s and enclosed mass $\approx 6\times10^{11}M_{\odot},$ even though higher resolution imaging data are needed to break residual degeneracies in model parameters. The longest time-delay (B-A) is estimated as $\approx 85$ (resp. $\approx125$) days by models with (resp. without) a perturber near G2/C. The configuration and predicted time-delays of J0408-5354 make it an excellent target for follow-up aimed at understanding the source quasar host galaxy and substructure in the lens, and measuring cosmological parameters. We also discuss some lessons learnt from J0408-5354 on lensed quasar finding strategies, due to its chromaticity and morphology.

Residual degeneracy from non-degenerate Landau levels of ultracold atoms in light-induced gauge potentials

We study non-degenerate Landau levels of ultracold trapped atoms in two dimensions, subject to an U(1)×U(1) Abelian gauge field and a lateral confining potential along a specific direction. The Landau-level degeneracy is removed due to the presence of the lateral confining potential that makes the single-particle energy spectrum explicitly dependent on the transverse momentum. The effect of the finite size of the atomic cloud on the energy spectrum is to split each Landau level into a set of sub-levels, once the transverse momentum becomes quantized. We show that under appropriate conditions some energy sub-levels overlap leading to a residual degeneracy of the system. Through numerical calculations, we map the residual degeneracy as a function of the effective magnetic field strength. Finally, we briefly discuss future studies on the transport properties of this atomic system that can be considered an optically induced atomic waveguide.

Quantum ergodicity for a class of non-generic systems

We examine quantum normal typicality and ergodicity properties for quantum systems whose dynamics are generated by Hamiltonians which have residual degeneracy in their spectrum and resonance in their energy gaps. Such systems can be considered atypical in the sense that degeneracy, which is usually a sign of symmetry, is naturally broken in typical systems due to stochastic perturbations. In particular, we prove a version of von Neumann’s quantum ergodic theorem, where a modified condition needs to hold in order to have normal typicality and ergodicity. As a result, we show that degeneracy of spectrum does not considerably modify the condition of the theorem, whereas the existence of resonance is more dominant for obstructing ergodicity.

The Structural, Anisotropic Magnetization, and Spectroscopic Study of Delafossite CuCr1−x M x O2 Systems

This work describes the chemical doping effect on the magnetic properties of p-type transparent semiconductor with delafossitetype oxides CuCr 0.95M0.05O2 (M = Ni, Cr, Ga, V, Fe, Mn, and Zn). The lattice parameters were found to vary according to Vegard’s law. Reflection broadening is observed, that is ascribed to local lattice distortion due to the ionic radius difference between Cr 3+ and M dopants Raman spectra at a room temperature of CuCr 0.95M0.05O2 displayed three intense modes which are identified as σ(A 1g) at 691 cm −1, σ(E g) at 444 cm −1 and σ(A g) at 207 cm −1 that matched with other delafossite structures. Magnetic susceptibility measurements show that the dominant interactions are antiferromagnetic (AFM) but that doping induces significant changes. A large ferromagnetic component arises in the Fe, V and Mn-substituted samples below ≈120 K. The Ni, Zn and Ga-substituted phases retain antiferromagnetic ordering at TN≈ 24 K. The main origin of the stabilization of the AFM state is probably the randomness induced by doping. The coupling between the local spins at the Cr sites and doped metal transition may enhance spin fluctuations at the Cr sites, which break the residual magnetic degeneracy as fluctuation-induced symmetry breaking in a highly magnetic degenerate groundstate manifold of some frustrated systems.

Mitigation of the LHC inverse problem

The LHC inverse problem refers to the difficulties in determining the parameters of an underlying theory from data (to be) taken by the LHC experiments: if they find signals of new physics, and an underlying theory is assumed, could its parameters be determined uniquely, or do different parameter choices give indistinguishable experimental signatures? This inverse problem was studied before for a supersymmetric Standard Model with 15 free parameters. This earlier study found 283 indistinguishable pairs of parameter choices, called degenerate pairs, even if backgrounds are ignored. We can resolve all but 23 of those pairs by constructing a true \chi^2 distribution using mostly counting observables. The elimination of systematic errors would even allow separating the residual degeneracies. Taking the Standard Model background into account we still can resolve 237 of the 283 "degenerate" pairs. This indicates that (some of) our observables should also be useful for the purpose of determining the values of SUSY parameters.

Phase diagram of the anisotropic multichannel Kondo Hamiltonian revisited

The phase diagram of the multichannel Kondo Hamiltonian with an $XXZ$ spin-exchange anisotropy is revisited, revealing a far richer fixed-point structure than previously appreciated. For a spin-$\frac{1}{2}$ impurity and $k>2$ conduction-electron channels, a second ferromagneticlike domain is found deep inside the antiferromagnetic regime. The new domain extends above a (typically large) critical longitudinal coupling ${J}_{z}^{*}>0$ and is separated from the antiferromagnetic domain by a second Kosterlitz-Thouless line. A similar line of stable ferromagneticlike fixed points with a residual isospin-$\frac{1}{2}$ local moment is shown to exist for large ${J}_{z}⪢\ensuremath{\mid}{J}_{\ensuremath{\perp}}\ensuremath{\mid}>0$ and arbitrary $k$ and $s$ obeying $\ensuremath{\mid}k\ensuremath{-}2s\ensuremath{\mid}>1$. Here, ${J}_{z}$ is the longitudinal spin-exchange coupling, ${J}_{\ensuremath{\perp}}$ is the transverse coupling, and $s$ is the impurity spin. Near the free-impurity fixed point, spin-exchange anisotropy generates a highly relevant term for $s>1∕2$ and arbitrary $k$. Depending on the sign of ${J}_{z}^{2}\ensuremath{-}{J}_{\ensuremath{\perp}}^{2}$ and the parity of $2s$, the system flows either to a conventional Fermi liquid with no residual degeneracy or to a $k$-channel, spin-$\frac{1}{2}$ Kondo effect or to a line of ferromagneticlike fixed points with a residual isospin-$\frac{1}{2}$ local moment. These results are obtained through a combination of perturbative renormalization-group techniques, Abelian bosonization, a strong-coupling expansion in $1∕{J}_{z}$, and explicit numerical renormalization-group calculations.

The ground state phases of orbitally degenerate spinel oxides

I review the microscopic spin–orbital Hamiltonian and ground state properties of spin one-half spinel oxides with threefold t 2 g orbital degeneracy. It is shown that for any orbital configuration a ground state of corresponding spin only Hamiltonian is infinitely degenerate in the classical limit. The extensive classical degeneracy is lifted by the quantum nature of the spins, an effect similar to order-out-of-disorder phenomenon by quantum fluctuations. This drives the system to a non-magnetic spin-singlet dimer manifold with a residual degeneracy due to relative orientation of dimers. The magneto-elastic mechanism of lifting the “orientational” degeneracy is also briefly reviewed.

Second-order effects for the hydrogen atom in perpendicular electric and magnetic fields and instability of classical electron trajectories

Degeneracy of energy levels of excited hydrogen atoms is not fully lifted by external perpendicular electric and magnetic fields in the first order of perturbation theory. Within the submanifold of residual degeneracy the levels splitting is governed by the second-order effects. The related classical electron trajectories exhibit instabilities characterized by the Lyapunov exponent. The present study establishes quantum mechanical implications of these instabilities. In particular, we derive from a quantum framework a simple analytical approximation for the Lyapunov exponent. The classical instabilities are linked to the presence of a diabatic quasistationary state in the quantum mechanical problem. Its meaning is elucidated by considering a related non-stationary quantum system with time-dependent field parameter. Some of the state-to-state transition probabilities are approximately expressed via Lyapunov exponents. Exact formulae for these probabilities are derived in the framework of multistate generalization of the Landau–Zener model. Non-adiabatic transitions are interpreted in terms of analytical properties of adiabatic potential curves.

Valence-bond crystal in a pyrochlore antiferromagnet with orbital degeneracy.

We discuss the ground state of a pyrochlore lattice of threefold orbitally degenerate S=1/2 magnetic ions. We derive an effective spin-orbital Hamiltonian and show that the orbital degrees of freedom can modulate the spin exchange, removing the infinite spin-degeneracy characteristic of pyrochlore structures. The resulting state is a collection of spin-singlet dimers, with a residual degeneracy due to their relative orientation. This latter is lifted by a magnetoelastic interaction, induced in the spin-singlet phase space, that forces a tetragonal distortion. Such a theory provides an explanation for the helical spin-singlet pattern observed in the B spinel MgTi2O4.

Broken time reversal symmetry in unconventional superconductors

Broken time reversal symmetry occurs in several unconventional superconductors at subdominant instabilities within the superconducting phase. The time reversal symmetry breaking state is, in general, stabilized through the additional opening of a gap. This is the case for both bulk and local instabilities at surfaces or interfaces. The time reversal symmetry breaking state in Sr 2RuO 4 is the residual degeneracy of the p-wave pairing state resulting from special properties of spin-orbit coupling.

Orbital ordering and strong correlations in manganites

We investigate the doped perovskite manganites using some elementary strongcorrelation models. We focus mainly on the electronic motion through doubleexchange and include the spin and lattice fluctuations solely to lift any residual degeneracy. We consider the lowenergy physics of the strongcoupling limit, incorporating the spatial structure of the orbitals carefully. Due to a mismatch between the pointsymmetries of the orbitals and the lattice, we find a highly geometrically frustrated Hubbardlike model. When orbital superexchange is dominant, the system finds ordering very difficult, and when the chargecarrier motion is dominant, the styles of orbital order are expected to be fairly complicated. Unless the system exhibits Fermiliquid behaviour, the strongcorrelations are suggested to be uncontrollable at present.

Two-laser multiphoton adiabatic passage in the frame of the Floquet theory. Applications to (1+1) and (2+1) STIRAP

We develop an adiabatic two-mode Floquet theory to analyse multiphoton coherent population transfer in N-level systems by two delayed laser pulses, which is a generalization of the three-state stimulated Raman adiabatic passage (STIRAP). The main point is that, under conditions of non-crossing and adiabaticity, the outcome and feasibility of a STIRAP process can be determined by the analysis of two features: (i) the lifting of degeneracy of dressed states at the beginning and at the end of the laser pulses, and (ii) the connectivity of these degeneracy-lifted branches in the quasienergy diagram. Both features can be determined by stationnary perturbation theory in the Floquet representation. As an illustration, we study the corrections to the RWA of the (1+1) STIRAP in strong fields and for large detunings. We analyse the possible breakdown of connectivity. In strong fields, the complete transfer is achieved, but the intermediate state, unpopulated within the RWA, can become populated during the process. In the (2+1) STIRAP, we show a residual degeneracy in a four-level system, that can be lifted by additional Stark shifts. The complete transfer is achieved under conditions of connectivity.

TOWARDS CONSTRUCTING ONE-PARTICLE REPRESENTATIONS OF THE DEFORMED POINCARÉ ALGEBRA

We give a method for obtaining states of massive particle representations of the two-parameter deformation of the Poincaré algebra proposed in Refs. 1–3. We discuss four procedures to generate eigenstates of a complete set of commuting operators starting from the rest state. One result of this work is the fact that upon deforming to the quantum Poincaré algebra, the rest state is split into an infinite number of states. Another result is that the energy spectrum of these states is discrete. Some curious residual degeneracy remains: there are states constructed by applying different operators to the rest state which nevertheless are indistinguishable by eigenvalues of all the observables in the algebra.

Photoabsorption spectra in the perturbative regime for atoms in crossed electric and magnetic fields

We calculate photoabsorption spectra of atoms in crossed electric and magnetic fields using a truncated basis of Coulomb eigenfunctions. The method yields spectra in the regime where inter-n-mixing is not dominant and allows for the treatment of non-hydrogenic atoms via a simple recourse to quantum defects. We compare results for hydrogen to those obtained in second order perturbation theory where the residual degeneracy left in first order perturbation theory is completely lifted and we show that only a very small basis size is needed to achieve convergence to within the accuracy of second order perturbation theory. In the case of lithium the coupling of an incomplete hydrogen-like manifold to states with non-negligible quantum defects substantially modifies the spectra obtained in comparison to the purely hydrogenic spectra. In the inter-n-mixing regime we also compare our convoluted results directly with an experimental spectrum for hydrogen and find good agreement below the saddle point.

MULTIPLE-Q STRUCTURES IN FRUSTRATED ANTIFERROMAGNETS

The concepts relevant to frustrated antiferromagnets are briefly reviewed. Antiferromagnets are classified according to their symmetry groups, with non-trivial point groups leading to the possibility of multiple-Q antiferromagnetism. The role of residual degeneracy is highlighted and the manner in which this degeneracy is lifted is discussed. The physical phenomena in competition within frustrated magnets, and the states that they prefer, yield ongoing theoretical research, and the way neutron scattering can be used, in conjunction with the application of pressure and magnetic fields, to determine which of the possible magnetic structures is stabilised is under experimental scrutiny. Multiple-Q antiferromagnetism finds varied and often exotic spin states with similar energies, and as such is the setting in which phase transitions between different magnetic states can be studied.

Accidental degeneracies in the zeeman effect and the symmetry groups

It is well known that the problem of a charged particle in a general central potential, subject to an external megnetic field, has no degeneracy. However, if the central potential has a symmetry group larger that O(3), as happens for the oscillator ( U(3)), Coulomb ( O(4)) and free particle ( E(3)), even in the presence of a magnetic field a residual degeneracy remains. In the present paper the symmetry Lie algebras are found which are responsible for the residual degeneracies in the problems mentioned, first in the case of weak magnetic fields when the Hamiltonian has only linear terms in the magnetic field intensity H and, in the last section, for strong magnetic fields when the term in H 2 is also included. Furthermore in the classical limit the canonical transformations are found that take the Hamiltonians of the problems mentioned in the presence of a magnetic field into other Hamiltonians in which the nature of the symmetry Lie algebra becomes obvious.

Removal of the degeneracy in the dual resonance model

We give a general formula which enables us to calculate the trajectory shift of any of the non-spurious resonance states caused by one loop perturbation. As immediate applications, the shift of the parent trajectory and the removal of two-fold degeneracy of the first daughter trajectory are explicity shown. Further, a suggestion is given that the one loop perturbation would completely remove the degeneracies of daughter trajectories and the residual degeneracies would be only accidental, even though the degree of degeneracy increases exponentially with mass.