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European Journal of Applied Sciences – Vol. 11, No. 3

Publication Date: June 25, 2023

DOI:10.14738/aivp.113.12083

Dandoloff, R. (2023). Knotted Configurations in the Continuous Heisenberg Spin Chain with Lower Bound of the Energy. European

Journal of Applied Sciences, Vol - 11(3). 825-828.

Services for Science and Education – United Kingdom

Knotted Configurations in the Continuous Heisenberg

Spin Chain with Lower Bound of theEnergy

R. Dandoloff

Dept. of Condensed Matter Physics and Microelectronics,

Faculty of Physics, Sofia University, 5 blvd. J. Bourchier, 1164 Sofia, Bulgaria

ABSTRACT

When studying the classical Heisenberg spin models, we usually map the

normalised unit vector,that represents the spin, on the tangent of a space curve.

The total chirality of the curve (the spin configuration) i.e., the total momentum

of the spin system, is a conserved quantity [1] and therefore self-crossing of the

curve is not allowed. Using this fact and the non-contractibility of some closed

loops in SO (3), we have proposed new topological spin configurations for the

Heisenberg spin model [1]. Now we propose new topological spin

configuration for the Heisenberg model, which represents a knot and has

higher lower bound for the energy as the previously proposed configurations.

Thesame analysis holds for a thin elastic rod that is bent to an open knot.

PACS numbers: 75.10. Pq, 75.10. Hk, 02.40. Hw, 02.10. Kn

Keywords: classical Heisenberg spin chain, knots, space curves

INTRODUCTION

In a recent paper [1] we have proposed new topological configurations for the one-dimensional

continuous Heisen-berg spin model wich have lower bound for the energy. In a following paper

[2] these configurations have been constructed as solutions of the Landau-Lifshitz equation

(nt = n × nss) where subscripts denote derivatives with respect of the arc length s and time

t. Our analysis is based on the classical theory of curves and the classical Hamiltonian mechanics

of the spin chain. The order parameter is normalised vector n (the spin) where n2 = 1. In angle

variables n = (sin θ cos ∅, sin θ sin ∅, cos θ). In these variables the Hamiltonian reads:

H = J ∫ (θs

2 + sin θ

2∅s

2

)ds +L

−L

(1)

where the subscript s stands for d/ds and s denotes the coordinate along R1 (the system is one- dimensional). The classical mechanics for this system has been worked out by Tjon and Wright

[3] . They have realised that Ø and cos θ can be considered as conjugated generalised coordinate

and momentum in which case the Poisson bracket reads [Ø (x), cos θ(y)] = δ (x − y). Then the

generator of translations (linear momentum) is:

P = ∫ (1 − cosθ)∅sds +L

−L

(2)

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Services for Science and Education – United Kingdom 826

European Journal of Applied Sciences (EJAS) Vol. 11, Issue 3, June-2023

It turns out that the linear momentum P is a constant of the motion. In order to use the

properties of space curves we will map the unit vector n to the unit tangent of a space curve [4],

so different space curves will represent different spin configurations. We will use homogeneous

boundary conditions where spins at ±L will be parallel.Then curves will tend to the straight line

as s → ±L. We are going touse the properties of the writhe of a curve (which characterises the

chirality of the curve). For a closed curve it is given by the following expression: [5]:

Wr =

1

4π

∮ ds ∮ ds′

(r(s)−r(s

′

)∙(n(s)−n(s

′

))

|r(s)−r(s

′)|

3

(3)

The tip of the radius vector r draws the curve, and n is its unit tangent. Using a theorem by

Fuller [6] one can express Wr with respect to a reference curve C0[7]:

Wr = Wr0 +

1

2π

∫

n0×n∙

d

ds(n0+n)

1+n0∙n

+L

−L

ds (4)

Here Wr0 is the writhe of the reference curve and after appropriate choice of the reference

curve we [7] get:

Wr =

1

2π

∫ (1 − cosθ)∅sds +L

−L

(5)

RESULTS AND DISCUSSION

Following the analysis in [1] we note that the write Wr coincides with the total momentum P

and hence the writhe W r is conserved quantity. We note also that when one region of the curve

crosses another one, the writhe W r suffers discontinuity by 2. Let us analyse now possible knot

configurations for the continuous one-dimensional classical Heisenberg spin chain. In [8] the

authors present a knot solution for the nonlinear Schrödinger equation (NLSE). They map the

NLSE onto the following equation for a space curve:

Rt = Rs × Rss (6)

Where the tip of the vector R (s, t) draws a non-stretching space curve. Here’s represents the

arc length of the space curve. Now let us take the derivative of the above equation with respect

to the arc length s and note that dR/ds = n, where n represents the tangent to the space curve

Rst = Rss × Rss + Rs × Rsss = Rs × Rsss

(7)

Here we recognise the Landau-Lifshits equation which turns out to be equivalent to the

NLSE.[9]

nt = n × nss (8)

n represents the normalised spin vector which has been mapped to the tangent of a space curve

that now represents a spin configuration. Now we are ready to examine the energy of the spin

configuration that represents a knot. It is well known that for a closed space curve [10]:

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Dandoloff, R. (2023). Knotted Configurations in the Continuous Heisenberg Spin Chain with Lower Bound of the Energy. European Journal of Applied

Sciences, Vol - 11(3). 825-828.

URL: http://dx.doi.org/10.14738/aivp.113.12083

∮ kds ≥ 2π (9)

We will use now yet another inequality for the integral curvature which concerns knots, namely

the Fary-Milnor theorem [11]:

∮ kds ≥ 4π (10)

The curvature is different from zero k 6= 0 only in the intervall s ∈ (−L, +L). Let us consider one

such configuration representing an open knot: the space curve representing the knot goes from

−L to +L. In order to get a closed curve this is completed by a straight line between −L and −∞

and between +L and +∞ and is closed by a semi-circle at infinity. The straight segments and the

semi-circle at infinity have curvature k zero. The writhe there is also 0. Then the above

inequality becomes:

∫ kds +L

−L

≥ 4π (11)

Let us note here that the Fary-Milnor inequality has been used to estimate the lower bound of

the number of localised electronic states on a trefoil knot [12].

Now we will apply the following Cauchy-Schwarz in equality to the integral curvature:

(∫ kds +L

−L

)

2

≤ 2L ∫ k

2ds +L

−L

(12)

If we introduce Euler angles the curvature is given by the following expression:

k

2 = θ s

2 + sin2 θ∅s

2

[13]

We are ready to calculate the lower bound of the energy for the spin chain with a knot

configuration. Now the energy of the spin chain satisfies the following inequality:

H = J ∫ (θs

2 + sin 2θ∅s

2

)ds +L

−L

= J ∫ k

2ds +L

−L

≥

J(∫ kds +L

−L

)

2

2L

≥ J

16 π

2

2L

= J

8 π

2

L

(13)

Here we note that the lower bound of the energy for the knot-configuration is 4 times higher

than the energy for the 2π- twist configuration studied in [1].

The same analysis can be applied to the case of a thin elastic rod that is bent/twisted into an

open knot. Its energy has the following lower bound (for the thin rod that is twisted by 2π and

whose end are glued together, see [14]):

E = Imin ∫ k

2ds +L

−L

≥ Imin

16 π

2

2L

= Imin

8 π

2

L

(14)

Where Imin stands for the minimal bending/twisting rigidity of the rod. In this case too the

lower bound of the energy of a knotted thin rod is 4 times higher than the lower bound for 2π

twisted rod.