Topological Properties of Phase Singularities in Wave Fields
Yi-Shi Duan, Ji-Rong Ren, and Tao Zhu∗
Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, P. R. China
(Dated: November 21, 2018)
Phase singularities as topological objects of wave fields appear in a variety of physical, chemical,
and biological scenarios. In this paper, by making use of the φ-mapping topological current theory,
we study the topological properties of the phase singularities in two and three dimensional space in
details. The topological inner structure of the phase singularities are obtained, and the topological
charge of the phase singularities are expressed by the topological numbers: Hopf indices and Brouwer
degrees. Furthermore, the topological invariant of the closed and knotted phase singularities in three
dimensional space are also discussed in details.
PACS numbers: 02. 40. Xx, 03. 65. Vf, 05. 45. Yv
I. INTRODUCTION
In physics, the solutions of wave equations in
two and three dimensional space often possess phase
singularities[1, 2]. At the point of phase singularity, the
phase of the wave is undefined and wave intensity van-
ishes. This phase singularity is a common property to
all waves, it exists in many area of physical, chemical,
and biological scenarios, such as the quantized vortices
in superfluid or superconductor systems[3, 4], the vor-
tices phase singularities in Bose-Einstein condensates[5],
the streamlines or singularities in quantum mechanical
wavefunctions[6, 7], the optical vortices in optical wave
systems[8, 9, 10, 11, 12, 13], the vortex filaments in chem-
ical reaction and molecular diffusion [15, 16, 17], and
the phase singularities in biological systems[17, 18]. In
recent years, the phase singularities have drawn great
interest because it is of importance for understanding
fundamental wave physics and have many important
applications, and a great deal of works on the phase
singularities in wave fields have been done by many
physicists[1, 2, 12, 13].
Phase singularities are generically points in two dimen-
sional space and form a network of lines in three di-
mensional space. Around the phase singularity points
in two dimensions or phase singularity lines in three
dimensions, the phase changes by 2π times an inte-
ger. This integer is the so-called topological charge (the
strength of the singularity) of the phase singularity ob-
jects (points or lines). In three dimensional space, a im-
portant case is that the phase singularity lines are closed
and knotted curves, these knot-like configurations exist
in a variety of physical scenarios, including Bose-Einstein
condensations[14], chemical reaction and molecular diffu-
sion systems[15, 16, 17], optical wave systems[19, 20] and
field theory[21, 22]. The phase singularities (points, lines,
or knotted) are topological objects, and have a very rich
topological properties, so it is necessary to use the topo-
logical method to study the topology of these objects.
∗Corresponding author. Email : zhut05@lzu.cn
In this paper, we have given a generic feature of the
topological properties of the phase singularities in wave
fields. By making use of the φ-mapping topological
current theory[23], we study the topological current of
the phase singularity objects (points in two dimensions
and lines in three dimensions), and the topological inner
structure of these topological objects are obtained. For
the case that the phase singularity lines in three dimen-
sional space are closed and knotted curves, we discussed
the topological invariant of these knotted family in de-
tails. This paper is arranged as follows. In Sec.II, we
construct a topological current density of the phase sin-
gularity points in two dimensions, this topological cur-
rent density don’t vanish only when the phase singular-
ity points exist, the topological charge of phase singu-
larity points are expressed by the topological quantum
numbers,the Hopf indices and Brouwer degrees of the φ-
mapping. In Sec.III is the topology of the phase singu-
larity lines in three dimensions. In Sec.IV, we introduce
a important topological invariant to describe the phase
singularity lines when they are linked and knotted, it is
just the sum of all the self-linking and all the linking
numbers of the knot family. In Sec.V is our concluding
remarks.
II. TOPOLOGICAL STRUCTURE OF PHASE
SINGULARITIES IN TWO DIMENSIONS
Let us study a complex scalar field ψ(~x, t), the wave
fields function ψ(~x, t) is maps from space (of either
two or three dimensions) to the complex numbers, so
ψ : R2, R3 → C. The complex scalar field ψ(~x, t) is time
dependent as well as space dependent, and it can be writ-
ten as
ψ(~x, t) = ‖ψ‖ eiθ = φ1(~x, t) + iφ2(~x, t), (1)
where φ1(~x, t) and φ2(~x, t) can be regarded as complex
representation of a two-dimensional vector field ~φ =
(φ1, φ2) over two or three dimensional space, ‖ψ‖ =√
ψ∗ψ is the modulus of ψ, and θ is the phase factor.
In this section, we will restrict our attention to the phase
http://arxiv.org/abs/0704.1190v1
singularities in two dimensional space, and labeled points
~x = (x1, x2) in cartesian coordinates. The phase singu-
larities in three dimensional space will be discussed in
Sec.III and Sec.IV.
It is known that the current density ~J associated with
ψ(~x, t) is defined as
~J = Im(ψ∗∇ψ) = ‖ψ‖2~v, (2)
the ~v is the velocity field. From the expressions in Eq.(1)
and Eq.(2), the velocity field ~v can be rewritten as
(ψ∗∇ψ −∇ψ∗ψ) = ∇θ, (3)
it becomes the gradient of the phase factor θ. The vor-
ticity field ~ω of the velocity field ~v is defined as
∇× ~v. (4)
Obviously, in two dimensional space the ~ω only has z-
component, i.e., ~ω = ω · ~ez. Form Eq.(3),we directly
obtain a trivial curl-free result: ~ω = 1
∇×∇θ = 0. But
in topology, because of the existence of phase singularities
in the wave fields ψ, the vorticity ~ω does not vanish[4].
So in the following discussions, we will study that what
the exact expression for ~ω is in topology.
Introducing the unit vector na = φa/‖φ‖(a =
1, 2;nana = 1), and defining a topological current in
(2+1) dimensions space-time,
ǫijkǫab∂jn
b, i = 0, 1, 2. (5)
Obviously, the topological current is identically con-
served,
i = 0. (6)
By making use of the definition of the unit vector na,
one can reexpressed the velocity field ~v as
~v = ǫabn
a∇nb, (7)
and the vorticity ~ω is
ω = j0 =
ǫjkǫab∂jn
b j, k = 1, 2, (8)
it is just the time component of the topological cur-
rent ji. According to the φ-mapping topological current
theory[23], one can prove that
ji = δ2(~φ)Di(
), (9)
where
εijkǫab∂jφ
b (10)
is the Jacobian vector, in which the time component
D0(φ/x) is the usual two dimensional Jacobian D(φ/x).
The expression of the topological current in Eq.(9) shows
that the topological current ji does not vanish only at
the zero points of ~φ, i.e., the phase singularities of the
wave fields ψ. According to the Eq.(8) and Eq.(9), the
vorticity can be expressed
ω = j0 = δ2(~φ)D(
), (11)
obviously, ω also does not vanish only when the phase
singularities of the wave fields ψ exist. The topological
current ji and vorticity ω paly an important role in the
topological property of the phase singularities, so it is
necessary to study the zero points of ~φ to determine the
non-zero solutions of ji.
According to the implicit function theory[24], while the
regular condition
Di(φ/x) 6= 0
is satisfied, the general solutions of
φ1(x1, x2, t) = 0, φ2(x1, x2, t) = 0, (12)
can be expressed as
x1 = x1k(s, t), x
2 = x2k(s, t), k = 1, 2, . . . , N, (13)
which represent N zero points ~zk (k=1, 2, . . . , N) where
ψ(~r, t) = 0 in two-dimensional space. These zero points
solutions are just the so-called phase singularity points
of the wave fields ψ, they represent the zeroes of the real
part and imaginary part of the complex scalar wave fields:
Re(ψ) = φ1 = 0 and Im(ψ) = φ2 = 0.
In the theory of δ function of ~φ(~r), one can prove that
in 2-dimensional space[25]
δ2(~φ) =
|D(φ/x)~zk |
δ2(~x− ~zk), (14)
where the positive integer βk is called the Hopf index of
map x→ φ[12, 13]. The meaning of βk is that when the
point ~zk covers the neighborhood Σk of zero ~zk once, the
vector field ~φ covers the corresponding region βk times.
By substituting Eq.(14) into Eq.(9), one can obtain that
2(~x− ~zk)
, (15)
whereWk = βkηk is the winding number of ~φ around zero
point ~zk, and ηk = sgn(D(φ/x)) = ±1 is the Brouwer
degrees of φ-mapping. The sign of Brouwer degrees are
very important, for the case of vortices phase singulari-
ties, the ηk = +1 corresponds to the vortex, and ηk = −1
corresponds to the antivortex.
The topological charge (the strength of phase singular-
ities) of the phase singularities point ~zk is defined by the
Gauss map ~n : ∂Σk → S1[26]:
n∗(ǫabn
adnb) (16)
where n∗ is the pullback of Gauss map n, ∂Σk is the
boundary of a neighborhood Σk of point ~zk and Σk ∩
Σm = ∅ for Σm is the neighborhood of another arbitrary
zero point ~zm. In topology it means that, when the point
~zk covers ∂Σk once, the unit vector ~n will cover S
1, or
~φ covers the corresponding region Wk times, which is a
topological invariant. By using the Stokes’ theorem and
in term of Eq.(8), one can obtain that
jk∂jn
ωd2x = βkηk. (17)
It is clear that the topological charges densities of phase
singularity points is just the non-vanishing vorticity ω of
the velocity field ~v. Using the notion j = (ω,~j), Eq.(6)
can be rewrite as
+∇ ·~j = 0, (18)
this expression tells us that the topological charges of the
phase singularity points are conserved. This is the only
topological property of the complex wave fields.
In this section, we obtain the topological inner struc-
ture of the phase singularity points in two dimensional
space, the solutions of the Eq.(13) possess N isolated
phase singularity points of which the kth point possesses
topological charge Wk = βkηk.
III. TOPOLOGICAL CURRENT OF PHASE
SINGULARITIES IN THREE DIMENSIONS
In this section,we will study the phase singularities in
three dimensions space, some results which discussed in
the above section in two dimensions space are also useful
here. Differently with in two dimensions, the point is
labeled ~x = (x1, x2, x3) and the vorticity ~ω have its all
three components. In three dimensional space, the ~ω can
be reexpressed as
ǫijkǫab∂jn
b, (19)
it is clearly that the vorticity ~ω is just the topological
current of phase singularities in three dimensions space.
Using the φ-mapping theory[23], the topological cur-
rent ~ω is rewritten as
ωi = δ2(~φ)Di(
), (20)
we can see from this expression that the vorticity ~ω is
non-vanishing only if ~φ = 0, i.e., the existence of the
phase singularities, so it is necessary to study these zero
solutions of ~φ. In three dimensions space, these solutions
are some isolate zero lines, which are the so-called phase
singularity lines in three dimensions space.
Under the regular condition
Di(φ/x) 6= 0,
the general solutions of
φ1(x1, x2, x3, t) = 0, φ2(x1, x2, x3, t) = 0 (21)
can be expressed as
x1 = x1k(s, t), x
2 = x2k(s, t), x
3 = x3k(s, t), (22)
which represent the world surfaces of N moving iso-
lated singular strings Lk with string parameter s (k =
1, 2, · · · , N). These singular strings solutions are just the
phase singularities solutions in three dimensions space.
In δ-function theory[25], one can obtain in three di-
mensions space
δ2(~φ) =
δ3(~x − ~xk()s)
ds, (23)
where
)|Σk =
ǫjkǫmn
and Σk is the kth planar element transverse to Lk with
local coordinates (u1, u2). The βk is the Hopf index of
φ mapping, which means that when ~x covers the neigh-
borhood of the zero point ~xk(s) once, the vector field
φ covers the corresponding region in φ space βk times.
Meanwhile the direction vector of Lk is given by
|xk =
Di(φ/x)
D(φ/u)
|xk . (24)
Then from Eq.(23) and Eq.(24) one can obtain the inner
structure of ωi:
δ3(~x− ~xk(s))ds, (25)
whereWk = βkηk is the winding number of ~φ around Lk,
with ηk = sgnD(φ/u)|~xk = ±1 being the Brouwer degree
of φ mapping. The sign of Brouwer degrees are very im-
portant, for the case of vortices phase singularities, the
ηk = +1 corresponds to the vortex, and ηk = −1 corre-
sponds to the antivortex. Hence the topological charge
of the phase singularity line Lk is[21]
ωidσi =Wk. (26)
The results in this section show us the topological inner
structure of the topological current density ωi. The topo-
logical charge of the kth phase singularity line in three
dimensions can be expressed by the topological numbers:
Qk =Wk = βkηk.
IV. TOPOLOGICAL ASPECT ON KNOTTED
PHASE SINGULARITY LINES
An important case is that the phase singularity lines in
three dimensions space from closed and knotted curves.
Topology has play a very important role in understanding
these knot configurations, so it is necessary to study the
topology in the knotted phase singularity lines. In order
to do that, we define a the helicity integral[27]
~v · ∇ × ~vd3x, (27)
this is an important topological knot invariant and it
measures the linking of the phase singularity lines. From
the Eq.(4), the helicity integral can be changed as
~v · ~ωd3x. (28)
Substituting Eq. (25) into Eq. (28), one can obtain
~v · d~x, (29)
when these phase singularities are closed and knotted
lines, i.e., a family of knots ξk(k = 1, 2, . . . , N), Eq. (29)
becomes
~v · d~x. (30)
It is well known that many important topological num-
bers are related to a knot family such as the self-linking
number and Gauss linking number. In order to discuss
these topological numbers of knotted phase singularity
lines, we define Gauss mapping:
~m : S1 × S1 → S2, (31)
where ~m is a unit vector
~m(~x, ~y) =
~y − ~x
|~y − ~x|
, (32)
where ~x and ~y are two points, respectively, on the knots
ξk and ξl (in particular, when ~x and ~y are the same point
on the same knot ξ , ~n is just the unit tangent vector ~T
of ξ at ~x ). Therefore, when ~x and ~y , respectively, cover
the closed curves ξk and ξl once, ~n becomes the section
of sphere bundle S2. So, on this S2 we can define the
two-dimensional unit vector ~e = ~e(~x, ~y). ~e, ~m are normal
to each other, i.e. ,
~e1 · ~e2 = ~e1 · ~m = ~e2 · ~m = 0,
~e1 · ~e1 = ~e2 · ~e2 = ~m · ~m = 1. (33)
In fact, the velocity field ~v can be decomposed in terms of
this two-dimensional unit vector ~e: vi = ǫabe
b − ∂iθ,
where θ is a phase factor[4]. Since one can see from the
expression ~ω = 1
∇ × ~v that the (∂iθ) term does not
contribute to the integral H , vi can in fact be expressed
vi = ǫabe
b. (34)
Substituting it into Eq.(14), one can obtain
a(~x, ~y)∂ie
b(~x, ~y)dxi. (35)
Noticing the symmetry between the points ~x and ~y in
Eq.(32), Eq.(35) should be reexpressed as
k,l=1
ǫab∂ie
bdxi ∧ dyj . (36)
In this expression there are three cases: (1) ξk and ξl are
two different phase singularities (ξk 6= ξl), and ~x and ~y
are therefore two different points (~x 6= ~y); (2) ξk and ξl
are the same phase singularities (ξk = ξl), but ~x and ~y
are two different points (~x 6= ~y); (3) ξk and ξl are the
same phase singularities (ξk = ξl), and ~x and ~y are the
same points (~x = ~y). Thus, Eq.(36) can be written as
three terms:
k=1(k=l, ~x 6=~y)
ǫab∂ie
bdxi ∧ dyj
k,l=1(k 6=l)
ǫab∂ie
bdxi ∧ dyj .(37)
By making use of the relation ǫab∂ie
b = 1
~m · (∂i ~m×
∂j ~m)[28], the Eq.(37) is just
k=1(~x 6=~y)
~m∗(dS)
k,l=1(k 6=l)
~m∗(dS), (38)
where ~m∗(dS) = ~m ·(∂i ~m×∂j ~m)dxi∧dyj(~x 6= ~y) denotes
the pullback of the S2 surface element.
In the following we will investigate the three terms in
the Eq.(38) in detail. Firstly, the first term of Eq.(38) is
just related to the writhing number[29] Wr(ξk) of ξk
Wr(ξk) =
~m∗(dS). (39)
For the second term, one can prove that it is related to
the twisting number Tw(ξk) of ξk
bdxi =
(~T × ~V ) · d~V
= Tw(ξk), (40)
where ~T is the unit tangent vector of knot ξk at ~x (~m = ~T
when ~x = ~y) and ~V is defined as ea = ǫabV b(~V ⊥ ~T ,~e =
~T × ~V ). In terms of the White formula[30]
SL(ξk) =Wr(ξk) + Tw(ξk), (41)
we see that the first and the second terms of Eq.(38) just
compose the self-linking numbers of knots.
Secondly, for the third term, one can prove that
~m∗(dS)
(xk − yk)
‖~x− ~y‖3
= Lk(ξk, ξl) (k 6= l), (42)
where Lk(ξk, ξl) is the Gauss linking number between ξk
and ξl[29]. Therefore, from Eqs.(39), (40), (41) and (42),
we obtain the important result:
W 2kSL(ξk) +
k,l=1(k 6=l)
WkWlLk(ξk, ξl). (43)
This precise expression just reveals the relationship be-
tween H and the self-linking and the linking numbers of
the phase singularity knots family[29]. Since the self-
linking and the linking numbers are both the invari-
ant characteristic numbers of the phase singularity knots
family in topology, H is an important topological invari-
ant required to describe the linked phase singularities in
wave fields.
V. CONCLUSION
In the present study, by making use of the φ-mapping
topological current theory, the topology of the phase sin-
gularity in wave fields is studied. Firstly, we obtain the
inner structure of the phase singularities in two and three
dimensional space. The phase singularity objects have
been found at the every zero point of the wave func-
tion ψ under the condition that the Jacobian determi-
nateDi(φ/x) 6= 0. One shows that the topological charge
(strength of the phase singularity) of the phase singular-
ity objects are determined by the topological numbers:
Hopf indices and Brouwer degrees. Secondly, we have
studied the topological invariant of the knotted phase sin-
gularity lines in three dimensional space in details. This
topological invariant can be expressed as the sum of all
the self-linking and all the linking numbers of the knotted
phase singularity lines.
Finally, it should be pointed out that in the present pa-
per, when we discussed the topological properties of the
phase singularities, the condition Di(φ/x) 6= 0 must be
satisfied. Now the question is coming, when this condi-
tion fails, what will happen about the phase singularity
objects? The answer is related to the evolution of the
phase singularity objects[31]. The topological object will
generate, annihilate, split, or merge when the condition
fails , and these dynamics properties of the phase singu-
larity objects will be discussed in further work.
Acknowledgments
This work was supported by the National Natural Sci-
ence Foundation of China.
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  Phase singularities as topological objects of wave fields appear in a variety
of physical, chemical, and biological scenarios. In this paper, by making use
of the $\phi$-mapping topological current theory, we study the topological
properties of the phase singularities in two and three dimensional space in
details. The topological inner structure of the phase singularities are
obtained, and the topological charge of the phase singularities are expressed
by the topological numbers: Hopf indices and Brouwer degrees. Furthermore, the
topological invariant of the closed and knotted phase singularities in three
dimensional space are also discussed in details.

Topological Properties of Phase Singularities in Wave Fields
Yi-Shi Duan, Ji-Rong Ren, and Tao Zhu∗
Institute of Theoretical Physics, Lanzhou University, Lanzhou 730000, P. R. China
(Dated: November 21, 2018)
Phase singularities as topological objects of wave fields appear in a variety of physical, chemical,
and biological scenarios. In this paper, by making use of the φ-mapping topological current theory,
we study the topological properties of the phase singularities in two and three dimensional space in
details. The topological inner structure of the phase singularities are obtained, and the topological
charge of the phase singularities are expressed by the topological numbers: Hopf indices and Brouwer
degrees. Furthermore, the topological invariant of the closed and knotted phase singularities in three
dimensional space are also discussed in details.
PACS numbers: 02. 40. Xx, 03. 65. Vf, 05. 45. Yv
I. INTRODUCTION
In physics, the solutions of wave equations in
two and three dimensional space often possess phase
singularities[1, 2]. At the point of phase singularity, the
phase of the wave is undefined and wave intensity van-
ishes. This phase singularity is a common property to
all waves, it exists in many area of physical, chemical,
and biological scenarios, such as the quantized vortices
in superfluid or superconductor systems[3, 4], the vor-
tices phase singularities in Bose-Einstein condensates[5],
the streamlines or singularities in quantum mechanical
wavefunctions[6, 7], the optical vortices in optical wave
systems[8, 9, 10, 11, 12, 13], the vortex filaments in chem-
ical reaction and molecular diffusion [15, 16, 17], and
the phase singularities in biological systems[17, 18]. In
recent years, the phase singularities have drawn great
interest because it is of importance for understanding
fundamental wave physics and have many important
applications, and a great deal of works on the phase
singularities in wave fields have been done by many
physicists[1, 2, 12, 13].
Phase singularities are generically points in two dimen-
sional space and form a network of lines in three di-
mensional space. Around the phase singularity points
in two dimensions or phase singularity lines in three
dimensions, the phase changes by 2π times an inte-
ger. This integer is the so-called topological charge (the
strength of the singularity) of the phase singularity ob-
jects (points or lines). In three dimensional space, a im-
portant case is that the phase singularity lines are closed
and knotted curves, these knot-like configurations exist
in a variety of physical scenarios, including Bose-Einstein
condensations[14], chemical reaction and molecular diffu-
sion systems[15, 16, 17], optical wave systems[19, 20] and
field theory[21, 22]. The phase singularities (points, lines,
or knotted) are topological objects, and have a very rich
topological properties, so it is necessary to use the topo-
logical method to study the topology of these objects.
∗Corresponding author. Email : zhut05@lzu.cn
In this paper, we have given a generic feature of the
topological properties of the phase singularities in wave
fields. By making use of the φ-mapping topological
current theory[23], we study the topological current of
the phase singularity objects (points in two dimensions
and lines in three dimensions), and the topological inner
structure of these topological objects are obtained. For
the case that the phase singularity lines in three dimen-
sional space are closed and knotted curves, we discussed
the topological invariant of these knotted family in de-
tails. This paper is arranged as follows. In Sec.II, we
construct a topological current density of the phase sin-
gularity points in two dimensions, this topological cur-
rent density don’t vanish only when the phase singular-
ity points exist, the topological charge of phase singu-
larity points are expressed by the topological quantum
numbers,the Hopf indices and Brouwer degrees of the φ-
mapping. In Sec.III is the topology of the phase singu-
larity lines in three dimensions. In Sec.IV, we introduce
a important topological invariant to describe the phase
singularity lines when they are linked and knotted, it is
just the sum of all the self-linking and all the linking
numbers of the knot family. In Sec.V is our concluding
remarks.
II. TOPOLOGICAL STRUCTURE OF PHASE
SINGULARITIES IN TWO DIMENSIONS
Let us study a complex scalar field ψ(~x, t), the wave
fields function ψ(~x, t) is maps from space (of either
two or three dimensions) to the complex numbers, so
ψ : R2, R3 → C. The complex scalar field ψ(~x, t) is time
dependent as well as space dependent, and it can be writ-
ten as
ψ(~x, t) = ‖ψ‖ eiθ = φ1(~x, t) + iφ2(~x, t), (1)
where φ1(~x, t) and φ2(~x, t) can be regarded as complex
representation of a two-dimensional vector field ~φ =
(φ1, φ2) over two or three dimensional space, ‖ψ‖ =√
ψ∗ψ is the modulus of ψ, and θ is the phase factor.
In this section, we will restrict our attention to the phase
http://arxiv.org/abs/0704.1190v1
singularities in two dimensional space, and labeled points
~x = (x1, x2) in cartesian coordinates. The phase singu-
larities in three dimensional space will be discussed in
Sec.III and Sec.IV.
It is known that the current density ~J associated with
ψ(~x, t) is defined as
~J = Im(ψ∗∇ψ) = ‖ψ‖2~v, (2)
the ~v is the velocity field. From the expressions in Eq.(1)
and Eq.(2), the velocity field ~v can be rewritten as
(ψ∗∇ψ −∇ψ∗ψ) = ∇θ, (3)
it becomes the gradient of the phase factor θ. The vor-
ticity field ~ω of the velocity field ~v is defined as
∇× ~v. (4)
Obviously, in two dimensional space the ~ω only has z-
component, i.e., ~ω = ω · ~ez. Form Eq.(3),we directly
obtain a trivial curl-free result: ~ω = 1
∇×∇θ = 0. But
in topology, because of the existence of phase singularities
in the wave fields ψ, the vorticity ~ω does not vanish[4].
So in the following discussions, we will study that what
the exact expression for ~ω is in topology.
Introducing the unit vector na = φa/‖φ‖(a =
1, 2;nana = 1), and defining a topological current in
(2+1) dimensions space-time,
ǫijkǫab∂jn
b, i = 0, 1, 2. (5)
Obviously, the topological current is identically con-
served,
i = 0. (6)
By making use of the definition of the unit vector na,
one can reexpressed the velocity field ~v as
~v = ǫabn
a∇nb, (7)
and the vorticity ~ω is
ω = j0 =
ǫjkǫab∂jn
b j, k = 1, 2, (8)
it is just the time component of the topological cur-
rent ji. According to the φ-mapping topological current
theory[23], one can prove that
ji = δ2(~φ)Di(
), (9)
where
εijkǫab∂jφ
b (10)
is the Jacobian vector, in which the time component
D0(φ/x) is the usual two dimensional Jacobian D(φ/x).
The expression of the topological current in Eq.(9) shows
that the topological current ji does not vanish only at
the zero points of ~φ, i.e., the phase singularities of the
wave fields ψ. According to the Eq.(8) and Eq.(9), the
vorticity can be expressed
ω = j0 = δ2(~φ)D(
), (11)
obviously, ω also does not vanish only when the phase
singularities of the wave fields ψ exist. The topological
current ji and vorticity ω paly an important role in the
topological property of the phase singularities, so it is
necessary to study the zero points of ~φ to determine the
non-zero solutions of ji.
According to the implicit function theory[24], while the
regular condition
Di(φ/x) 6= 0
is satisfied, the general solutions of
φ1(x1, x2, t) = 0, φ2(x1, x2, t) = 0, (12)
can be expressed as
x1 = x1k(s, t), x
2 = x2k(s, t), k = 1, 2, . . . , N, (13)
which represent N zero points ~zk (k=1, 2, . . . , N) where
ψ(~r, t) = 0 in two-dimensional space. These zero points
solutions are just the so-called phase singularity points
of the wave fields ψ, they represent the zeroes of the real
part and imaginary part of the complex scalar wave fields:
Re(ψ) = φ1 = 0 and Im(ψ) = φ2 = 0.
In the theory of δ function of ~φ(~r), one can prove that
in 2-dimensional space[25]
δ2(~φ) =
|D(φ/x)~zk |
δ2(~x− ~zk), (14)
where the positive integer βk is called the Hopf index of
map x→ φ[12, 13]. The meaning of βk is that when the
point ~zk covers the neighborhood Σk of zero ~zk once, the
vector field ~φ covers the corresponding region βk times.
By substituting Eq.(14) into Eq.(9), one can obtain that
2(~x− ~zk)
, (15)
whereWk = βkηk is the winding number of ~φ around zero
point ~zk, and ηk = sgn(D(φ/x)) = ±1 is the Brouwer
degrees of φ-mapping. The sign of Brouwer degrees are
very important, for the case of vortices phase singulari-
ties, the ηk = +1 corresponds to the vortex, and ηk = −1
corresponds to the antivortex.
The topological charge (the strength of phase singular-
ities) of the phase singularities point ~zk is defined by the
Gauss map ~n : ∂Σk → S1[26]:
n∗(ǫabn
adnb) (16)
where n∗ is the pullback of Gauss map n, ∂Σk is the
boundary of a neighborhood Σk of point ~zk and Σk ∩
Σm = ∅ for Σm is the neighborhood of another arbitrary
zero point ~zm. In topology it means that, when the point
~zk covers ∂Σk once, the unit vector ~n will cover S
1, or
~φ covers the corresponding region Wk times, which is a
topological invariant. By using the Stokes’ theorem and
in term of Eq.(8), one can obtain that
jk∂jn
ωd2x = βkηk. (17)
It is clear that the topological charges densities of phase
singularity points is just the non-vanishing vorticity ω of
the velocity field ~v. Using the notion j = (ω,~j), Eq.(6)
can be rewrite as
+∇ ·~j = 0, (18)
this expression tells us that the topological charges of the
phase singularity points are conserved. This is the only
topological property of the complex wave fields.
In this section, we obtain the topological inner struc-
ture of the phase singularity points in two dimensional
space, the solutions of the Eq.(13) possess N isolated
phase singularity points of which the kth point possesses
topological charge Wk = βkηk.
III. TOPOLOGICAL CURRENT OF PHASE
SINGULARITIES IN THREE DIMENSIONS
In this section,we will study the phase singularities in
three dimensions space, some results which discussed in
the above section in two dimensions space are also useful
here. Differently with in two dimensions, the point is
labeled ~x = (x1, x2, x3) and the vorticity ~ω have its all
three components. In three dimensional space, the ~ω can
be reexpressed as
ǫijkǫab∂jn
b, (19)
it is clearly that the vorticity ~ω is just the topological
current of phase singularities in three dimensions space.
Using the φ-mapping theory[23], the topological cur-
rent ~ω is rewritten as
ωi = δ2(~φ)Di(
), (20)
we can see from this expression that the vorticity ~ω is
non-vanishing only if ~φ = 0, i.e., the existence of the
phase singularities, so it is necessary to study these zero
solutions of ~φ. In three dimensions space, these solutions
are some isolate zero lines, which are the so-called phase
singularity lines in three dimensions space.
Under the regular condition
Di(φ/x) 6= 0,
the general solutions of
φ1(x1, x2, x3, t) = 0, φ2(x1, x2, x3, t) = 0 (21)
can be expressed as
x1 = x1k(s, t), x
2 = x2k(s, t), x
3 = x3k(s, t), (22)
which represent the world surfaces of N moving iso-
lated singular strings Lk with string parameter s (k =
1, 2, · · · , N). These singular strings solutions are just the
phase singularities solutions in three dimensions space.
In δ-function theory[25], one can obtain in three di-
mensions space
δ2(~φ) =
δ3(~x − ~xk()s)
ds, (23)
where
)|Σk =
ǫjkǫmn
and Σk is the kth planar element transverse to Lk with
local coordinates (u1, u2). The βk is the Hopf index of
φ mapping, which means that when ~x covers the neigh-
borhood of the zero point ~xk(s) once, the vector field
φ covers the corresponding region in φ space βk times.
Meanwhile the direction vector of Lk is given by
|xk =
Di(φ/x)
D(φ/u)
|xk . (24)
Then from Eq.(23) and Eq.(24) one can obtain the inner
structure of ωi:
δ3(~x− ~xk(s))ds, (25)
whereWk = βkηk is the winding number of ~φ around Lk,
with ηk = sgnD(φ/u)|~xk = ±1 being the Brouwer degree
of φ mapping. The sign of Brouwer degrees are very im-
portant, for the case of vortices phase singularities, the
ηk = +1 corresponds to the vortex, and ηk = −1 corre-
sponds to the antivortex. Hence the topological charge
of the phase singularity line Lk is[21]
ωidσi =Wk. (26)
The results in this section show us the topological inner
structure of the topological current density ωi. The topo-
logical charge of the kth phase singularity line in three
dimensions can be expressed by the topological numbers:
Qk =Wk = βkηk.
IV. TOPOLOGICAL ASPECT ON KNOTTED
PHASE SINGULARITY LINES
An important case is that the phase singularity lines in
three dimensions space from closed and knotted curves.
Topology has play a very important role in understanding
these knot configurations, so it is necessary to study the
topology in the knotted phase singularity lines. In order
to do that, we define a the helicity integral[27]
~v · ∇ × ~vd3x, (27)
this is an important topological knot invariant and it
measures the linking of the phase singularity lines. From
the Eq.(4), the helicity integral can be changed as
~v · ~ωd3x. (28)
Substituting Eq. (25) into Eq. (28), one can obtain
~v · d~x, (29)
when these phase singularities are closed and knotted
lines, i.e., a family of knots ξk(k = 1, 2, . . . , N), Eq. (29)
becomes
~v · d~x. (30)
It is well known that many important topological num-
bers are related to a knot family such as the self-linking
number and Gauss linking number. In order to discuss
these topological numbers of knotted phase singularity
lines, we define Gauss mapping:
~m : S1 × S1 → S2, (31)
where ~m is a unit vector
~m(~x, ~y) =
~y − ~x
|~y − ~x|
, (32)
where ~x and ~y are two points, respectively, on the knots
ξk and ξl (in particular, when ~x and ~y are the same point
on the same knot ξ , ~n is just the unit tangent vector ~T
of ξ at ~x ). Therefore, when ~x and ~y , respectively, cover
the closed curves ξk and ξl once, ~n becomes the section
of sphere bundle S2. So, on this S2 we can define the
two-dimensional unit vector ~e = ~e(~x, ~y). ~e, ~m are normal
to each other, i.e. ,
~e1 · ~e2 = ~e1 · ~m = ~e2 · ~m = 0,
~e1 · ~e1 = ~e2 · ~e2 = ~m · ~m = 1. (33)
In fact, the velocity field ~v can be decomposed in terms of
this two-dimensional unit vector ~e: vi = ǫabe
b − ∂iθ,
where θ is a phase factor[4]. Since one can see from the
expression ~ω = 1
∇ × ~v that the (∂iθ) term does not
contribute to the integral H , vi can in fact be expressed
vi = ǫabe
b. (34)
Substituting it into Eq.(14), one can obtain
a(~x, ~y)∂ie
b(~x, ~y)dxi. (35)
Noticing the symmetry between the points ~x and ~y in
Eq.(32), Eq.(35) should be reexpressed as
k,l=1
ǫab∂ie
bdxi ∧ dyj . (36)
In this expression there are three cases: (1) ξk and ξl are
two different phase singularities (ξk 6= ξl), and ~x and ~y
are therefore two different points (~x 6= ~y); (2) ξk and ξl
are the same phase singularities (ξk = ξl), but ~x and ~y
are two different points (~x 6= ~y); (3) ξk and ξl are the
same phase singularities (ξk = ξl), and ~x and ~y are the
same points (~x = ~y). Thus, Eq.(36) can be written as
three terms:
k=1(k=l, ~x 6=~y)
ǫab∂ie
bdxi ∧ dyj
k,l=1(k 6=l)
ǫab∂ie
bdxi ∧ dyj .(37)
By making use of the relation ǫab∂ie
b = 1
~m · (∂i ~m×
∂j ~m)[28], the Eq.(37) is just
k=1(~x 6=~y)
~m∗(dS)
k,l=1(k 6=l)
~m∗(dS), (38)
where ~m∗(dS) = ~m ·(∂i ~m×∂j ~m)dxi∧dyj(~x 6= ~y) denotes
the pullback of the S2 surface element.
In the following we will investigate the three terms in
the Eq.(38) in detail. Firstly, the first term of Eq.(38) is
just related to the writhing number[29] Wr(ξk) of ξk
Wr(ξk) =
~m∗(dS). (39)
For the second term, one can prove that it is related to
the twisting number Tw(ξk) of ξk
bdxi =
(~T × ~V ) · d~V
= Tw(ξk), (40)
where ~T is the unit tangent vector of knot ξk at ~x (~m = ~T
when ~x = ~y) and ~V is defined as ea = ǫabV b(~V ⊥ ~T ,~e =
~T × ~V ). In terms of the White formula[30]
SL(ξk) =Wr(ξk) + Tw(ξk), (41)
we see that the first and the second terms of Eq.(38) just
compose the self-linking numbers of knots.
Secondly, for the third term, one can prove that
~m∗(dS)
(xk − yk)
‖~x− ~y‖3
= Lk(ξk, ξl) (k 6= l), (42)
where Lk(ξk, ξl) is the Gauss linking number between ξk
and ξl[29]. Therefore, from Eqs.(39), (40), (41) and (42),
we obtain the important result:
W 2kSL(ξk) +
k,l=1(k 6=l)
WkWlLk(ξk, ξl). (43)
This precise expression just reveals the relationship be-
tween H and the self-linking and the linking numbers of
the phase singularity knots family[29]. Since the self-
linking and the linking numbers are both the invari-
ant characteristic numbers of the phase singularity knots
family in topology, H is an important topological invari-
ant required to describe the linked phase singularities in
wave fields.
V. CONCLUSION
In the present study, by making use of the φ-mapping
topological current theory, the topology of the phase sin-
gularity in wave fields is studied. Firstly, we obtain the
inner structure of the phase singularities in two and three
dimensional space. The phase singularity objects have
been found at the every zero point of the wave func-
tion ψ under the condition that the Jacobian determi-
nateDi(φ/x) 6= 0. One shows that the topological charge
(strength of the phase singularity) of the phase singular-
ity objects are determined by the topological numbers:
Hopf indices and Brouwer degrees. Secondly, we have
studied the topological invariant of the knotted phase sin-
gularity lines in three dimensional space in details. This
topological invariant can be expressed as the sum of all
the self-linking and all the linking numbers of the knotted
phase singularity lines.
Finally, it should be pointed out that in the present pa-
per, when we discussed the topological properties of the
phase singularities, the condition Di(φ/x) 6= 0 must be
satisfied. Now the question is coming, when this condi-
tion fails, what will happen about the phase singularity
objects? The answer is related to the evolution of the
phase singularity objects[31]. The topological object will
generate, annihilate, split, or merge when the condition
fails , and these dynamics properties of the phase singu-
larity objects will be discussed in further work.
Acknowledgments
This work was supported by the National Natural Sci-
ence Foundation of China.
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