The Multifaceted Skyrmion

The following sections are included:

• Preface
• Contents
• Introduction

We review recent work on the modelling of atomic nuclei as quantised 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 for baryon numbers which are multiples of four, that 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 quantisation 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 Skyrme model has two Skyrmion solutions of baryon number 12, with D_3h and D_4h symmetries. The first has an equilateral triangular shape and the second an extended linear shape, analogous to the triangle and linear chain structures of three alpha particles. We recalculate the moments of inertia of these Skyrmions, and deduce the energies and spins of their quantised rotational excitations. There is a good match with the ground-state band of Carbon-12, and with the recently established rotational band of the Hoyle state. The ratio of the root mean square matter radii also matches the experimental value.

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 colors 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 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.

Using the bound state version of the topological soliton model for the baryons we show that the existence of a bound (or quasi-bound) $\overline{D}$-soliton state leads to the possibility of having hidden charm pentaquarks with quantum numbers and masses, which are compatible with those of the candidates recently reported by the LHCb experiment. The implications of heavy quark symmetry are elaborated.

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 f_π at a density n_1/2 lower than or near the critical density n_c at which hadronic matter changes over to a chiral symmetry restored phase with possibly deconfined quarks. When hidden local gauge fields and a dilaton scalar of spontaneously broken scale symmetry with decay constant f_χ are incorporated, this half-skyrmion phase is characterized by f_χ ≈ f_π ≠ 0 with the hidden gauge coupling g ≠ 0 but ≪ 1. While chiral symmetry is restored globally in this region in the sense that space-averaged, vanishes, quarks are still confined in massive hadrons and massless pions. This phase is shown to play a crucial role in the model for a smooth transition from a soft EoS at low density to a stiffer EoS at high density, the changeover taking place at n_1/2. It resembles the “quarkyonic phase” predicted in large N_c QCD and represents the “hadronic freedom” regime which figures as a doorway to chiral restoration. The fractionization of skyrmion matter into half-skyrmion matter has a tantalizing analogy to what appears to happen in condensed matter in (2+1) dimensions where half-skyrmions or “meron” enter as relevant degrees of freedom at the interface.

In this review, we summarize the main features of the BPS Skyrme model which provides a physically well-motivated idealization of atomic nuclei and nuclear matter: (1) it leads to zero binding energies for classical solitons (while realistic binding energies emerge owing to the semiclassical corrections, the Coulomb interaction and isospin breaking); (2) it describes a perfect non-barotropic fluid already at the microscopic (field theoretical) level which allows to study thermodynamics beyond the mean-field limit. These properties allow for an approximate but analytical calculation of binding energies of the most abundant nuclei, for a determination of the equation of state of skyrmionic matter (both in the full field theory and in a mean-field approximation) as well as the description of neutron stars as Skyrme solitons with a very good agreement with available observational data.

All these results suggest that the proper low energy effective model of QCD should be close to the BPS Skyrme model in a certain sense (a "near-BPS Skyrme model"), with a prominent role played by the BPS part.

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 history of modern condensed matter physics may be regarded as the competition and reconciliation between Stoner's and Anderson's physical pictures, where the former is based on momentum-space descriptions focusing on long wave-length fluctuations while the latter is based on real-space physics emphasizing emergent localized excitations. In particular, these two view points compete with each other in various nonperturbative phenomena, which range from the problem of high T_c superconductivity, quantum spin liquids in organic materials and frustrated spin systems, heavy-fermion quantum criticality, metal-insulator transitions in correlated electron systems such as doped silicons and two-dimensional electron systems, the fractional quantum Hall effect, to the recently discussed Fe-based superconductors. An approach to reconcile these competing frameworks is to introduce topologically nontrivial excitations into the Stoner's description, which appear to be localized in either space or time and sometimes both, where scattering between itinerant electrons and topological excitations such as skyrmions, vortices, various forms of instantons, emergent magnetic monopoles, and etc. may catch nonperturbative local physics beyond the Stoner's paradigm. In this review article we discuss nonperturbative effects of topological excitations on dynamics of correlated electrons. First, we focus on the problem of scattering between itinerant fermions and topological excitations in antiferromagnetic doped Mott insulators, expected to be relevant for the pseudogap phase of high T_c cuprates. We propose that nonperturbative effects of topological excitations can be incorporated within the perturbative framework, where an enhanced global symmetry with a topological term plays an essential role. In the second part, we go on to discuss the subject of symmetry protected topological states in a largely similar light. While we do not introduce itinerant fermions here, the nonperturbative dynamics of topological excitations is again seen to be crucial in classifying topologically nontrivial gapped systems. We point to some hidden links between several effective field theories with topological terms, starting with one dimensional physics, and subsequently finding natural generalizations to higher dimensions.

Quantum mechanics is a strange business, and the quantum physics of strongly correlated many-electron systems can be stranger still. Good examples are the various quantum Hall effects. They are among the most remarkable many-body quantum phenomena discovered in the second half of the 20th century, comparable in intellectual import to superconductivity and superfluidity. The quantum Hall effects are an extremely rich set of phenomena with deep and truly fundamental theoretical implications…

Charged excitations in quantum Hall (QH) systems are noncommutative skyrmions. QH systems represent 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 applying a certain unitary transformation to an electron or hole 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.

Bilayer two-dimensional electron gas systems can form unusual broken symmetry states with spontaneous inter-layer phase coherence at certain filling factors. At total filling factor ν_T = 1, the lowest energy charged excitation of the system is theoretically suggested to be a linearly-confined meron-pair, which is topologically identical to a single skyrmion. We will review how this remarkable excitation arises and can help unravel various experimental results demonstrated in bilayer quantum Hall system. In order to detect the linearly-confined meron-pair excitation directly, we propose a gated bilayer Hall bar experiment, where the magnitude and orientation of magnetic field B_‖ applied parallel to the 2D plane can be controlled. We demonstrate a strong angle-dependent transport due to the anisotropic nature of linearly-confined meron-pairs and discuss how it would be manifested in experiment.

The following sections are included:

• Introduction
• Microscopic Theory for Skyrmions
• The Skyrme Crystal
• Collective Modes and Quantum Fluctuations
• The Bilayer Quantum Hall System
• Conclusion
• Acknowledgments
• 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.

Broken symmetry states characterizing density waves of higher angular momentum in correlated electronic systems are intriguing objects. In the scheme of characterization by angular momentum, conventional charge and spin density waves correspond to zero angular momentum. Here we explore a class of exotic density wave states that have topological properties observed in recently discovered topological insulators. These rich topological density wave states deserve closer attention in not only high temperature superconductors but in other correlated electron states, as in heavy fermions, of which an explicit example will be discussed. The state discussed has non-trivial charge 2e skyrmionic spin texture. These skyrmions can condense into a charged superfluid. Alternately, they can fractionalize into merons and anti-merons. The fractionalized particles that are confined in skyrmions in the insulating phase, can emerge at a deconfined quantum critical point, which separates the insulating and the superconducting phases. These fractional particles form a two-component spin-singlet chiral (d_x²−y² ± id_xy) wave superconducting state that breaks time reversal symmetry. Possible connections of this exotic order to the superconducting state in the heavy-fermion material URu₂Si₂ are suggested. The direct evidence of such a chiral superconducting state is polar Kerr effect that was observed recently.

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.

Skyrmions are topological solitons that describe baryons within a nonlinear theory of pions. In holographic QCD, baryons correspond to topological solitons in a bulk theory with an extra spatial dimension: thus the three-dimensional Skyrmion lifts to a four-dimensional holographic Skyrmion in the bulk. We begin this review with a description of the simplest example of this correspondence, where the holographic Skyrmion is exactly the self-dual Yang-Mills instanton in flat space. This places an old result of Atiyah and Manton within a holographic framework and reveals that the associated Skyrme model extends the nonlinear pion theory to include an infinite tower of vector mesons, with specific couplings for a BPS theory. We then describe the more complicated curved space version that arises from the string theory construction of Sakai and Sugimoto. The basic concepts remain the same but the technical difficulty increases as the holographic Skyrmion is a curved space version of the Yang-Mills instanton, so self-duality and integrability are lost. Finally, we turn to a low-dimensional analogue of holographic Skyrmions, where aspects such as multi-baryons and finite baryon density are amenable to both numerical computation and an approximate analytic treatment.

In a wide class of holographic models, like the one proposed by Sakai and Sugimoto, baryons can be approximated by instantons of non-abelian gauge fields that live on the world-volume of flavor D-branes. In the leading order, those are just the Yang-Mills instantons, whose solutions can be constructed from the celebrated ADHM construction. This fact can be used to study various properties of baryons in the holographic limit. In particular, one can attempt to construct a holographic description of the cold dense nuclear matter phase of baryons. It can be argued that holographic baryons in such a regime are necessarily in a solid crystalline phase. In this review we summarize the known results on the construction and phases of crystals of the holographic baryons.

The following sections are included:

• Author Index
• Subject Index