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Bulletin of PFUR Series Mathematics. Information Sciences. Physics. No 2 (2). 2010. Pp. 152-156
Физика
UDC 517.957, 530.145
On Two-Field Solitons in 2 and 3 Dimensions I. L. Bogolubsky, A. A. Bogolubskaya
Laboratory of Information Technologies Joint Institute for Nuclear Research Joliot-Curie 6, 141980 Dubna, Moscow region, Russia
We study two- and three-dimensional stationary solitons with non-trivial topology in gauge-invariant nonlinear sigma models (NSMs) describing interaction of scalar unit SN fields with gauge vector SU(N - 1) fields, N =2, 3.
Key words and phrases: nonlinear sigma models, solitons.
1. Introduction
Nonlinear sigma models (NSMs) are of great importance in the modern mathematical physics, which is due to their universality: they appear in various branches of fundamental science. Classical NSMs describe evolution in time of ^-component unit isovector field sa(x, t) in (D + 1)-dimensional space-time (a = 1,... ,N + 1); field manifolds of these models are unit spheres SN. The most interesting cases correspond to D = 2,3 and N = 2, 3.
Below we discuss the A3M model with (N = 2,D = 2), introduced in [1], and the A4YM model with (N = 3,D = 3), introduced in [2]. The A4YM model is the straightforward extension of the A3M model. On the other hand, one can see deep resemblance of the A4YM model with the bosonic sector of the reduced electroweak Salam-Weinberg theory, widely known as SU2-Higgs model, in which radial degree of freedom of the Higgs field is frozen (see the A4YM Lagrangian in Sec. 2). In fact, our gauged NSMs include: i) unit length scalar (N + 1)-component field, with values on SN : si2 + ...sw+i2 = 1 (N = 2, 3), interacting with ii) vector field with U(1) or SU(2) symmetry (Maxwell or Yang-Mills).
2. A3M model in 2 dimensions
Consider minimal interaction of the S2 scalar field (A3-field) with the Maxwell field A^(x). The resulting "A3M model" is described by the gauge-invariant Lagrangian:
£ = + d^s3d^s3 - V(sa) - 4Fl„,
"¿V = d/j. + ieAn, Vn = d^ - ieAp, (1)
S+ = S1 + IS2, S- = S1 — IS2,
F{1V = d{1Av - dvAv, v(sa) = p(1 - si),
where are coupling constants and ¡i,v = 0,1,2. The localized distributions of unit isovector sa(x) in this model are divided into classes with different topological inidices ("charges") Qt; solitons with nonzero topological charges are referred to as
Received 28th November, 2009.
"topological solitons" [1]. We look for the topological solitons of the A3M model using the "hedgehog-like" ansatz for the A3-field
si = cos mx sin $(R), s2 = sin mx sin $(R), s3 = cos $(R),
y ^ t~>2 2 2
sin x = ^ cos x = ^ R = x + y ,
where m is an integer number. We use also the standard "vortex" ansatz for the vector field A^, describing localized distributions of a stationary magnetic field:
V X
^o = 0, Ai = Ax = -ma(R) , M = Ay = ma(R) For them Qt = m.
After rescaling (a = ae-1, R = re-1,) we calculate 5H/5$ and 5H/5a, arriving at coupled equations for $(r) and a(r)
d2 •& 1 d$
—T- H------sin v cos V
dr 2 r dr
to2 (a — 1)2
+ p
0, (2)
d2a 1 da . 2 *
-r-2 —-"T" + 2sin2tf(1 - a) = 0, (3)
dr2 r dr
to be solved under the following boundary conditions:
0(0) = -K, ê(œ) = 0, (4)
a(0) = 0, ^ (to) = 0. (5)
dr
Using series expansion of ê(r) and a(r) at r ^ 0, we find from Eqs. (2) and (3) for to = 1
ê(r) = n — C\r + o(r),
a(r) = r2(^ El — 1 Cy^j + o(r4),
and for to = 2
■d(r) = n — C2r2 + o(r2),
a(r) = r2( £2 — ^ ^r4) + o(r6).
The asymptotic form of the soliton solution for r ^ to is:
T
$(r) ~ —¡= exp(-^fpr), T = const,
V^ T2
a(r) « a^ - (1 - a^)-— exp(-2^/pr).
2rp
We studied the problem (2)-(5) by various numerical methods, among them shooting technique, stabilization method. The method based on power and asymptotic series and on the analytic continuation technique (re-expansions and Pade approximants) was used as well [3].
Solutions exist and are stable for the values of dimensionless anisotropy parameter
0 <p <pcr « 0.41.
The plots of radial functions a(r) and $(r) and corresponding distributions of energy density and magnetic field have been presented in [1]. Later we have found the dependence of Qt = 1 soliton energy on parameter p [4]. It is interesting to note
that the dependence «(to) on p proved to be surprisingly symmetric (see Fig. 1). Presently the only way to explain such a symmetry is to refer to high (U(1) <8> Z(2)) symmetry of the A3M model (1). Then we studied Qt = 2 solitons. We have found that for all 0 < p < pcr ~ 0.41 their energies turned out to satisfy inequality Esoi(Qt = 2,p) < 2 * Esoi(Qt = 1,p). This means that two Qt = 1 solitons attract to each other, forming the Qt = 2 bound states as a result of initial configuration evolution.
0 1 0.2 0 3 0.4
Figure 1. «(to) vs p for the ASM model.
3. A4YM model for D = 3
Further we shall consider another gauged sigma model, which describe minimal interaction of the easy-axis 4-component unit isovector field qa(x("the A4-field") interacting with the vector SU(2) Yang-Mills field Aa^(xv). The Lagrangian density of this ("the A4YM") model is:
L = V^Wq* + d,q0d^q0 - V(q0) - )2, V„qa = d^qa + geabcAy, Fl„ = d,Al - dvA1 + gs^A^Al, V(q0) = p [1 - (q0)2] ,
where = 0,1, 2, 3; a,b,c =1,2, 3; fi,g are coupling constants.
First we looked for stationary topological solitons of the A4YM model using the following ansatz for the A4- and the SU(2) Yang-Mills fields:
q0 = cos 0(R), qa = sin §(R) ^, R2 = + y2 + z2,
R
= 0, A:f = c(R)eiak xk.
Then the Hamiltonian density distributions of localized field bunches are spherically symmetric:
Hst(R) = (2 + + 4gcsin2r& + 2g2c2R2sin2V + 6c2+
dR J R2
+ (M) R2 + ^g2c4R4 + 4R-cT~n + 2gR2c3 + /3sin2§.
?2 , 1 2 4D4 .at, „dc , o„ T>2„3 . o- 2„
2* dR
Introduce dimensionless variables r = gR, b(r) = g 1cr2
tions ( P =
Calculating 5%/5§ and 5%/5r, we get coupled equations ^P = -^j
d2§ 2 dV
-¡-^ H------sin § cos v
d 2 d
M HP
d2 2 2 ^ - ^ - 2sin2§(l + 6) - b-(6 + 3) = 0.
(6)
When searching for localized solutions we set the following boundary conditions: 0(0) = k, tf(ro) = 0, 6(0) = 0, b{<x>) = G. (7)
Solutions to above problem (6)-(7) would define localized distributions qa(xk) ( a = 0,1, 2, 3, and k = 1,2, 3) of the A4-field with unit topological charge, Qt = 1. Here Qt is the "mapping degree" of continuous maps R?comp ^ S3. However such solutions have not been found. Because of that we look for more general ansatz.
More general ansatz keeps the "hedgehog" form for qa and a generalized expression for Ai:
R +naUl R QA?
However such ansatz should respect Lorentz gauge. Equating ——^ = 0, we find
OXi
At(x) = eaijUjC(R)R + (5ai - nam+ nam—^. (8)
B(R) = - 2dR (R2D(R)). (9)
Finally we obtain the ansatz defined by equations (8) and (9).
We calculate the hamiltonian density %st(R) for such ansatz using the computer algebra system Maple [5]. Equatmg var.at.onal der.vat.ves , »
oC oD ov to 0, we obtain coupled equations for radial functions C(R),D(R),V(R). Their solutions (if exist) define localized soliton solutions to A4YM model. The study of coupled equations for C(R),D(R),§(R) is in progress.
0
4. Conclusions
In this paper we discussed the existence and properties of localized solutions of the A3M model (D = 2) and the A4YM model (D = 3). Topological solitons of these models can be considered as soliton analogues of the so-called defect solutions: 2D strings-vortices in the Abelian Higgs model [6,7] and 3 D't Hooft-Polyakov "hedgehogs"-monopoles [8-10] correspondingly.
References
1. Bogolubsky I. L., Bogolubskaya A. A. 2D Topological Solitons in the Gauged Easy-Axis Heisenberg Antiferromagnet Model // Phys. Lett. B. — 1997. — Vol. 395. — Pp. 269-274.
2. Bogolubsky I. L., Bogolubskaya A. A. On Multidimensional Topological Solitons in Gauged Sigma Models with Spontaneously Broken Z(2) Symmetry // Ann. Fond. Louis de Broglie. — 1998. — Vol. 23, No 1. — Pp. 11-14.
3. Bogolubsky A. I., Skorokhodov S. L. Pade Approximants, Symbolic Evaluations, and Computation of Solitons in Two-Field Antiferromagnet Model // Programming and Computer Software. — 2004. — Vol. 30, No 2. — Pp. 95-99.
4. Stable-Extended String-Vortex Solitons / I. L. Bogolubsky, A. A. Bogolubskaya, A. I. Bogolubsky, S. L. Skorokhodov // Path Integrals - New Trends and Perspectives. Proceedings of the 9th International Conference / Ed. by W. Janke, A. Peltser. — Singapore: World Scientific, 2008.
5. Maple 12 User Manual. — http://www.maplesoft.com/view.aspx?sid=5883.
6. Abrikosov A. A. On the Magnetic Properties of Superconductors of the Second Group // Soviet Physics JETP. — 1957. — Vol. 5, No 6. — Pp. 1174-1182.
7. Nielsen H. B., Olesen P. Vortex-Line Models for Dual Strings // Nuclear Physics. — 1973. — Vol. B61. — Pp. 45-61.
8. 't Hooft G. Magnetic Monopoles in Unified Gauge Theories // Nuclear Physics. — 1974. — Vol. B79. — Pp. 276-284.
9. Polyakov A. M. Particle Spectrum in Quantum Field Theory // JETP Lett. — 1974. — Vol. 20, No 6. — Pp. 194-195.
10. Polyakov A. M. Isomeric States of Quantum Fields // JETP. — 1975. — Vol. 68. — Pp. 1975-1990.
УДК 517.957, 530.145
Двухполевые солитоны в двух- и трехмерном пространстве
И. Л. Боголюбский, А. А. Боголюбская
Лаборатория информационных технологий Объединенный институт ядерных исследований ул. Жолио-Кюри д.6, Дубна, Московская область, 141980, Россия
В работе исследуются двух- и трехмерные стационарные солитоны с нетривиальной топологией в калибровочно-инвариантных нелинейных сигма-моделях, описывающих взаимодействие скалярных полей со значениями на сферах с калибровочными векторными ви(Ж — 1) полями (Ж = 2, 3).
Ключевые слова: нелинейные сигма-модели, солитоны.
