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UDC 517.957, 530.145 On 2D and 3D Localized Solutions with Nontrivial Topology
I. L. Bogolubsky, A. A. Bogolubskaya
Laboratory of Information Technologies Joint Institute for Nuclear Research 6, Joliot-Curie str., Dubna, Moscow region, Russia, 141980
Localized solutions of nonlinear field models with nontrivial topological properties are discussed. Existence of various systems of definitions of the topological objects, developed in this area of research historically, can potentially lead to the wrong conclusions about existence of such solutions. The classification allowing to define accurately and differentiate objects with different topological properties is proposed, which prevents from inferring wrong conclusions. Such classification is especially important for multidimensional solutions. Such solutions are divided into 2 classes: the topological solitons (TS) and topological defects (TD). Solutions of both types describe the localized distributions of field energy, but they differ in topological properties. We exemplify and compare stationary TSs and TDs in 2 and 3 spatial dimensions. Examples of TSs are: solitons in Heisenberg magnets, Belavin-Polyakov solitons/instantons, Skyrmions, "baby-skyrmions". Examples of TDs are: sine-Gordon kinks, Nielsen-Olesen strings-vortices in the Abelian Higgs (AHM) model, 't Hooft-Polyakov hedgehog-monopoles in the Georgi-Glashow model. We note some technical problems with TDs, which are not met in the case of TSs. Soliton analogs of Nielsen-Olesen TDs in the AHM have been found: they are TSs in the A3M model. We have started search for TSs in the SU(2)-Higgs model which is currently in progress.
Key words and phrases: topological charge, solitons, defects, mapping degree, Abelian-Higgs model, Yang-Mills field, Heisenberg antiferromagnet, Georgi-Glashow and Weinberg-Salam models, hedgehog ansatz.
1. Introduction
Investigation of localized energy distributions described by solutions of nonlinear field equations with nontrivial topological properties is the important approach to nonperturbative field theory. Here we propose a classification of localized topological solutions (often called topological lumps, TL) which differ from standard one but seems to be more instructive, in particular when studying new multidimensional (D = 2,3) particle-like solutions with topological charges.
Historically the first localized solutions with nontrivial topology were skyrmions, found in [1] and used for the description of baryons. In our classification they are topological solitons (TS). More than 10 years later great interest has been drawn to the 2D Nielsen-Olesen strings-vortices [2] and to the 3D't Hooft-Polyakov hedgehog-monopoles [3-5]; these solutions - in our classification - belong to topological defects (TD). Note that the above strings-vortices and hedgehog-monopoles have been discovered during the great solitonic boom of 70's, and that is why they were ascribed to the wide class of solitons (lumps). However we believe that the usage of the same term "a soliton" both for TSs and TDs may turn out misleading in some cases (for an example see Sect.6, where TLs in the Standard Model are discussed).
2. Definitions
Both topological defects, TD, and topological solitons, TS, describe particle-like (extended localized, lumps) distributions of field energy, but they (TDs and TSs) differ
Received 22nd September, 2013.
in topological properties. For TSs field distributions in R° of all fields involved are uniform at space infinity, R ^ to (see for example Fig. 1, where space distribution of 3-component unit Heisenberg field in magnetic soliton is depicted). In this paper consider SN-valued field with N = D; then for TSs topological charge (index) is a mapping degree of the SN —valued field distribution inside infinite radius (R = to) sphere SD-1, which is considered — because of constancy of all fields on it — as the single point. The space R° is compactified by adding this infinite point, and thus soliton maps R^omp ^ &N.
Figure 1. TS with Qtop = 1 in Heisenberg magnet (reproduced from [6])
Contrary to TSs Topological Defects are given by SN-valued field distributions, which are nonuniform at R = to. Their topological indices are mapping degrees of the R = to sphere SD-1 to a SN sphere defined by the field distribution on this sphere, SD-1 ^ SN ,N = D — 1. From above definitions it is clear that Topological Defects are not Topological Solitons, and vice versa, Topological Solitons are not Topological Defects.
3. Examples of Topological Solitons and Defects
Now we present some examples of TSs and TDs in D=2 and D=3.
1. Topological Solitons: solitons in Heisenberg magnets (2D, 3D) [6,7], Belavin-Polyakov solitons/instantons (2D) [8,9], Skyrmions (3D) [1], "baby-skyrmions" (20), see, e.g. [10].
2. Topological Defects: sine-Gordon kinks (10), Nielsen-Olesen strings-vortices in the Abelian Higgs model (2D) [2], 't Hooft-Polyakov hedgehog-monopoles in the Georgi-Glashow model (3D) [3].
4. Drawbacks of Topological Defects
We believe that TDs have some drawbacks, which are connected with nonuni-formity of SN-valued field at R = to (see Fig. 2a). The first of them is that they cannot be generated from unperturbed vacuum state in a finite time. The second one is a problem of matching two (or more) defects. It can be clearly illustrated in two-dimensional (D = 2) case for S^valued field. In fact, consider two well-separated defects with unit topological charge (see Fig. 2b), so that the center of the first one is located on the £-axis at x1 = —to, and the centre of the second one at x2 = +to. Then from Fig. 2b one can see that in the vicinity of vertical line x = 0 it is impossible to define distribution of S*1 field which is consistent with asymptotic behavior of both defects in this vicinity. To circumvent the second problem one can either insert "junctions" in between defects or consider multidefects configurations. Both ways seems to be not quite satisfactory.
In the case of TSs one doesn't encounter such difficulties. That is why we are interested in search of solitonic analogs of topological defects both in D = 2 and D = 3 cases; in another words, it is interesting and important to find soliton analogs
Bogolubsky I. L., Bogolubskaya A.A. On 2D and 3D Localized Solutions .
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of Nielsen-Olesen strings-vortices in D = 2 and of 't Hooft-Polyakov monopoles-hedgehogs in D = 3.
Figure 2. Left: Isolated defect solution, D = 2, S^-valued field. Right: Problems
with matching two D = 2 defects
5. Topological Solitons in the A3M model
Instead of complex scalar field in the Abelian Higgs model (AHM) in the A3M model [11] we introduce 3-component unit isovector scalar field sa(x) taking values on unit sphere S2 : sasa = 1, a = 1, 2, 3, having selfinteraction of so-called "easy-axis" type (well-known in magnetism theory). Similar to AHM introduce gauge-invariant interaction of this field with the Maxwell field, making global U(1) symmetry of easy-axis magnets local one. As a results we arrive at "the A3M model", first introduced and studied in [11]. The Lagrangian density of the A3M-model is
£ = V^S-V»s+ + dßs3d»S3 - V(sa) - 4F^, = dß + ieAß, = dß - ieAß,
S+ = Si + IS2, S- = Si — IS2, FpV = d^A, - dvA^, V(sa) = ß(1 - sl),
(1)
where ¡i,v = 0,1,..., D. This model is the gauge-invariant extension of the classical Heisenberg antiferromagnet model with the easy-axis anisotropy. It supports D = 2 topological solitons, which can be found using the "hedgehog" ansatz for the unit isovector field s^(x), i = 1,2,3,
s1 = cos mx sin $(R), S2 = sin mx sin $(R), s3 = cos $(R),
y X d2 2 , 2
sin x = ^ cos x = ^ R = x + y,
(2)
where m is an integer number, and the "vortex" ansatz for the Maxwell field A^(x),
v x
Aq = 0, Ai = Ax = -ma(R)^-, A2 = Ay = ma(R)-—.
R2'
R?
(3)
The topological charge of A3M solitons is defined as the mapping degree of sa(x) distributions inside infinite radius (R = to) sphere, ^Comp ^ &2. Boundary conditions correspond to uniform distribution of the sa(x) field at R = to, and zero value of the Maxwell field A^ (x) at space infinity. A3M solitons exist [11] for integer Qtop — similar to Belavin-Polyakov 2D solitons in isotropic Heisenberg magnet.
Energy of two A3M solitons with Qtop = 1 proves to be greater than energy of one soliton with Qtop = 2 [11]. As a result two such solitons attract to each other and coalesce into one Qtop = 2 soliton. Thus soliton analogs of Nielsen-Olesen TDs in the AHM have been found.
6. Topological Solitons in the SU2-Higgs Model
Consider the simplest electroweak (EW) model (a reduction of the bosonic sector of the Weinberg-Salam model), the so-called SU2-Higgs model with
L = (V^$)t(V^$) - 1 Fa»v - ($t$ - i)2, (4)
V^b = + %-gTaAl$bl ¡j, = 0, i, 2, 3, a = i, 2, 3, b = i, 2, $ is the 2-component complex isospinor, defined by 4 real numbers <pc, such that (pc(pc = 1, c = 1,2,3,4. Introduce unit isospinor field <1 = $/\J$t$ and (pa = (pa/\/'-pc'-pc, a = 1, 2, 3,4 so that normalized field (pa takes values on unit sphere S3. The SU2-Higgs model describes gauge-invariant interaction of SU(2) Yang-Mills field with the isospinor scalar field. Let boundary conditions at R = to, R? = x2 + y2 + z2: be <Pc(to) = (p0, i.e. (p0 = (0, 0, 0,1), or <pg = (0,0,0, —1). The topological charge Qtop can be defined as the mapping degree of R?comp ^ S3 given by distribution of the 4-component unit field (pa(x) inside infinite radius sphere R = to.
Existence of topological solitons with integer topological charge Qtop is not a priori excluded. To find TSs one can use
(i) hedgehog ansatz for isospinor field with chosen Qtop, in the simplest case Qtop — 1 it takes the form
<p4 = cos ^(r) ■ f (r), Cp3 = sin^(r) ■ cos^(r) ■ f (r), Cp2 = sin^(r) ■ sin •Q(r) ■ sin ip(x, y) ■ f (r), (5)
(pi = sin^(r) ■ sin$(r) ■ cos ip(x, y) ■ f(r), sin^ = y/\/ (x2 + y2), cos^ = x/\/ (x2 + y2),
here ip(r), 'd(r) and f(r) are to be found by minimization of topological lump energy and
(ii) generic 3-term ansatz for D = 3 Yang-Mills solitons (Aa = 0):
g Ai - e iak R2 S(R) + —R3T
K 7)2 \ | P( R)xixa
(5iaRZ - XiXa) +
R4
(6)
i,k - 1,2,3, R2 -x2 + y2 + z2.
Study of Topological Solitons in the SU(2)-Higgs model is in progress.
Note that the SU(2)-Higgs model does not support TDs, because the 4-component unit field defined on the sphere S2 has no nontrivial topological properties. This however does not mean that there is no possibility for existence of TSs in this model because maps R3comp ^ S3 are divided into classes with different integer topological charges. This example shows that usage of the generic term soliton for TDs and TSs can lead to erroneous conclusion.
References
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УДК 517.957, 530.145 О двумерных и трёхмерных локализованных решениях с нетривиальной топологией
И. Л. Боголюбский, А. А. Боголюбская
Лаборатория информационных технологий Объединённый институт ядерных исследований ул. Жолио-Кюри, д. 6, Дубна, Московская область, Россия, 141980
Обсуждаются частицеподобные решения нелинейных полевых моделей с нетривиальными топологическими свойствами. Существование различных систем определений топологических объектов, сложившееся в этой области исследований исторически, может приводить к неправильным выводам о существовании подобных решений. Предлагается классификация, позволяющая чётко определить и разграничить объекты разными топологическими свойствами, что помогает избежать ошибочных заключений. Такая классификация особенно важна для многомерных решений. Естественным образом можно выделить 2 класса таких решений: топологические солитоны (ТС) и топологические дефекты (ТД). И те, и другие описывают частицеподобные распределения полевой энергии, но различаются по топологическим свойствам. Приводятся примеры и проводится сравнение ТС и ТД для случаев 2 и 3 пространственных измерений. В рамках выбранной системы определений к классу ТС можно отнести солитоны в Гейзенбергских магнетиках, солитоны-инстантоны Белавина—Полякова, скирмионы, «бэби-скирмионы». К классу ТД—кинки синус-уравнения Гордона, вихри-струны Нильсена-Ольсена в Абе-левой модели Хиггса (АМХ), ежи-монополи т'Хуфта—Полякова в модели Джорджи— Глэшоу. Отмечается, что при работе с ТД возникают некоторые технические проблемы; в случае ТС таких трудностей не возникает. Описывается солитонный аналог дефектов Нильсена-Ольсена в АМХ — это топологические солитоны в АЗМ-модели. Сформулирована задача поиска топологических солитонов в 8и(2)-модели Хиггса; их поиск продолжается.
Ключевые слова: топологический заряд, солитоны, дефекты, степень отображения, абелева модель Хиггса, поле Янга—Миллса, антиферромагнетик Гейзенберга, модели Джорджи-Глэшоу и Вайнберга—Салама, «ежовая» подстановка.
