The Multifaceted Skyrmion

The following sections are included:

• Preface

• Contents

• Introduction

We review recent work on the modelling of atomic nuclei as quantized Skyrmions, using Skyrme's original model with pion fields only. Skyrmions are topological soliton solutions whose conserved topological charge B is identified with the baryon number of a nucleus. Apart from an energy and length scale, the Skyrme model has just one dimensionless parameter m proportional to the pion mass. It has been found that a good fit to experimental nuclear data requires m to be of order 1. The Skyrmions for B up to 7 have been known for some time, and are qualitatively insensitive to whether m is zero or of order 1. However, for baryon numbers B = 8 and above, the Skyrmions have quite a compact structure for m of order 1, rather than the hollow polyhedral structure found when m = 0. One finds that for baryon numbers which are multiples of four, the Skyrmions are composed of B = 4 sub-units, as in the α-particle model of nuclei.

The rational map ansatz gives a useful approximation to the Skyrmion solutions for all baryon numbers when m = 0. For m of order 1, it gives a good approximation for baryon numbers up to 7, and generalisations of this ansatz are helpful for higher baryon numbers.

We briefly review the work from the 1980s and 90s on the semiclassical rigidbody quantization of Skyrmions for B = 1, 2, 3 and 4. We then discuss more recent work extending this method to B = 6, 7, 8, 10 and 12. We determine the quantum states of the Skyrmions, finding their spins, isospins and parities, and compare with the experimental data on the ground and excited states of nuclei up to mass number 12.

The ratio of electric to magnetic proton form factors as measured in polarization transfer experiments shows a characteristic linear decrease with increasing momentum transfer Q²(< 10 (GeV/c)²). We present a simple argument how such a decrease arises naturally in chiral soliton models. For a detailed comparison of model results with experimentally determined form factors it is necessary to employ a boost from the soliton rest frame to the Breit frame. To enforce asymptotic counting rules for form factors, the model must be supplemented by suitably chosen interpolating powers n in the boost prescription. Within the minimal π-ϱ-ω soliton model, with the same n for both, electric and magnetic form factors, it is possible to obtain a very satisfactory fit to all available proton data for the magnetic form factor and to the recent polarization results for the ratio . At the same time the small and very sensitive neutron electric form factor is reasonably well reproduced. The results show a systematic discrepancy with presently available data for the neutron magnetic form factor for Q² > 1 (GeV/c)². We additionally comment on the possibility to extract information about the form factors in the time-like region and on two-photon exchange contributions to unpolarized elastic scattering which specifically arise in soliton models.

We outline how one can understand the Skyrme model from the modern perspective. We review the quantization of the SU(3) rotations of the Skyrmion, leading to the exotic baryons that cannot be made of three quarks. It is shown that in the limit of large number of colours the lowest-mass exotic baryons can be studied from the kaon-Skyrmion scattering amplitudes, an approach known after Callan and Klebanov. We follow this approach and find, both analytically and numerically, a strong Θ⁺ resonance in the scattering amplitude that is traced to the rotational mode. The Skyrme model does predict an exotic resonance Θ⁺ but grossly overestimates the width. To understand better the factors affecting the width, it is computed by several different methods giving, however, identical results. In particular, we show that insofar as the width is small, it can be found from the transition axial constant. The physics leading to a narrow Θ⁺ resonance is briefly reviewed and affirmed.

The description of the heavy baryons as heavy-meson–soliton bound systems is reviewed. We outline how such bound systems arise from effective lagrangians that respect both chiral symmetry and heavy quark symmetry. Effects due to finite heavy quark masses are also discussed, and the resulting heavy baryon spectra are compared with existing quark model and empirical results. Finally, we address some issues related to a possible connection between the usual bound state approach to strange hyperons and that for heavier baryons.

We review an approach, developed over the past few years, to describe hadronic matter at finite density and temperature, whose underlying theoretical framework is the Skyrme model, an effective low energy theory rooted in large N_c QCD. In this approach matter is described by various crystal structures of skyrmions, classical topological solitons carrying baryon number, from which conventional baryons appear by quantization. Chiral and scale symmetries play a crucial role in the dynamics as described by pion, dilaton and vector meson degrees of freedom. When compressed or heated skyrmion matter describes a rich phase diagram which has strong connections with the confinement/deconfinement phase transition.

The hadronic matter described as a skyrmion matter embedded in an FCC crystal is found to turn into a half-skyrmion matter with vanishing (in the chiral limit) quark condensate and non-vanishing pion decay constant at a density n_½S lower than or at the critical density at which hadronic matter changes over to a chiral symmetry restored phase with deconfined quarks. When hidden local gauge fields and dilaton scalars — one “soft” and one “hard” — are incorporated, this phase is characterized by a = 1, ƒ_π≠ 0 with the hidden gauge coupling g ≠ 0 but ≪ 1. While chiral symmetry is restored in this region in the sense that , quarks are still confined in massive hadrons and massless pions. This phase seems to correspond to the “quarkyonic phase” predicted in large N_c QCD. It also represents the “hadronic freedom” regime relevant to kaon condensation at compact-star density. As g → 0 (in the chiral limit), the symmetry “swells” — as an emergent symmetry due to medium — to SU(N_ƒ)⁴ as proposed by Georgi for the “vector limit.” The fractionization of skyrmion matter into halfskyrmion matter is analogous to what appears to happen in condensed matter in (2+1) dimensions where half-skyrmions or “merons” enter as relevant degrees of freedom at the interface. Finally the transition from baryonic matter to color-flavor- locked quark matter can be bridged by a half-skyrmion matter.

QCD predicts matter at high density should exhibit color superconductivity. We review briefly several pertinent properties of color superconductivity and then discuss how baryons are realized in color superconductors. Especially, we explain an attempt to describe the color-flavor locked quark matter in terms of bosonic degrees of freedom, where the gapped quarks and Fermi sea are realized as Skyrmions, called superqualitons, and Q-matter, respectively.

We discuss one of the most interesting phenomena exhibited by baby skyrmions — breaking of rotational symmetry. The topics we will deal with here include the appearance of rotational symmetry breaking in the static solutions of baby Skyrme models, both in flat as well as in curved spaces, the zero-temperature crystalline structure of baby skyrmions, and finally, the appearance of spontaneous breaking of rotational symmetry in rotating baby skyrmions.

The following sections are included:

• Quantum Hall phenomena

• Quantum Hall ferromagnetism

• Skyrmions

• Isospin Ordering in Bilayer Systems

• Other examples of pseudospin order

• References

Charged excitations in quantum Hall (QH) systems are noncommutative skyrmions. QH systems present an ideal system equipped with noncommutative geometry. When an electron is confined within the lowest Landau level, its position is described solely by the guiding center, whose X and Y coordinates do not commute with one another. Topological excitations in such a noncommutative plane are noncommutative skyrmions flipping several spins coherently. We construct a microscopic skyrmion state by making a certain unitary transformation of a hole or electron state. A remarkable property is that a noncommutative skyrmion carries necessarily the electron number proportional to the topological charge. More remarkable is the bilayer QH system with the layer degree of freedom acting as the pseudospin, where the quasiparticle is a topological soliton to be identified with the pseudospin skyrmion. Such a skyrmion is deformed into a bimeron (a pair of merons) by the parallel magnetic field penetrated between the two layers. Each meron carries the electric charge ±e/2.

The bilayer two-dimensional electron gas systems can form an unusual broken symmetry state with spontaneous inter-layer phase coherence at certain filling factors. At total filling factor v_T = 1, the lowest energy charged excitation of the system is proposed to be a meron-pair, which is topologically identical to a single skyrmion. We will review how this remarkable excitation arises theoretically and can help unravel the novel experimental results of bilayer quantum Hall system.

The following sections are included:

• Introduction

• Microscopic Theory for Skyrmions

• The Skyrme Crystal

• Collective Modes and Quantum Fluctuations

• The Bilayer Quantum Hall System
  • Two-dimensional superfluidity and Josephson physics
  • The coherence network model
  • Effect of interlayer bias

• Conclusion

• Acknowledgements

• References

We review the half-Skyrmion theory for copper-oxide high-temperature superconductivity. In the theory, doped holes create a half-Skyrmion spin texture which is characterized by a topological charge. The formation of the half-Skyrmion is described in the single hole doped system, and then the half-Skyrmion excitation spectrum is compared with the angle-resolved photoemission spectroscopy results in the undoped system. Multi-half-Skyrmion configurations are studied by numerical simulations. We show that half-Skyrmions carry non-vanishing topological charge density below a critical hole doping concentration ˜ 30% even in the absence of antiferromagnetic long-range order. The magnetic structure factor exhibits incommensurate peaks in stripe ordered configurations of half-Skyrmions and anti-half-Skyrmions. The interaction mediated by half-Skyrmions leads to d_x²−y²-wave superconductivity. We also describe pseudogap behavior arising from the excitation spectrum of a composite particle of a half-Skyrmion and doped hole.

The theory of second-order phase transitions is one of the foundations of modern statistical mechanics and condensed-matter theory. A central concept is the observable order parameter, whose nonzero average value characterizes one or more phases. At large distances and long times, fluctuations of the order parameter(s) are described by a continuum field theory, and these dominate the physics near such phase transitions. We show that near second-order quantum phase transitions, subtle quantum interference effects can invalidate this paradigm, and we present a theory of quantum critical points in a variety of experimentally relevant two-dimensional antiferromagnets. The critical points separate phases characterized by conventional “confining” order parameters. Nevertheless, the critical theory contains an emergent gauge field and “deconfined” degrees of freedom associated with fractionalization of the order parameters. We propose that this paradigm for quantum criticality may be the key to resolving a number of experimental puzzles in correlated electron systems and offer a new perspective on the properties of complex materials.

We review recent progress in baryon physics using gauge/string duality. Skyrme's idea to realize baryons as solitons is beautifully embedded in string theory.

We review baryons in the D4-D8 holographic model of low energy QCD, with the large N_c and the large 't Hooft coupling limit. The baryon is identified with a bulk soliton of a unit Pontryagin number, which from the four-dimensional viewpoint translates to a modified Skyrmion dressed by condensates of spin one mesons. We explore classical properties and find that the baryon in the holographic limit is amenable to an effective field theory description. We also present a simple method to capture all leading and subleading interactions in the 1/N_c and the derivative expansions. An infinitely predictive model of baryon-meson interactions is thus derived, although one may trust results only for low energy processes, given various approximations in the bulk. We showcase a few comparisons to experiments, such as the leading axial couplings to pions, the leading vector-like coupling, and a qualitative prediction of the electromagnetic vector dominance that involves the entire tower of vector mesons.

The Cheshire cat principle states that hadronic observables at low energy do not distinguish between hard (quark) or soft (meson) constituents. As a result, the delineation between hard/soft (bag radius) is like the Cheshire cat smile in Alice in Wonderland. This principle reemerges from current holographic descriptions of chiral baryons whereby the smile appears in the holographic direction. We illustrate this point for the baryonic form factor.

We review the procedure to calculate baryonic properties using a recently proposed five-dimensional approach to QCD. We show that this method gives predictions to baryon observables that agree reasonably well with the experimental data.

The following sections are included:

• Author Index

• Subject Index