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Moving discrete curve and geometric phase
20 May 1996
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 2 14 ( 1996) 252-258
Moving discrete curve and geometric phase Masato Hisakado, Miki Wadati Depurtment
of Physics, Faculty
of Science, University
ofTokyo,
7-3-I
Hongo,
1 November 1995; revised manuscript
Received
received 25 January 1996; accepted Communicated by A.R. Bishop
Bunkyo-ku,
Tokyo 113. Japun
for publication
26 February
1996
Abstract The study of space curves finds many applications ih physics, such as optical fibers, magnetic spin chains, and vortex filaments in a fluid. The time evolution of a space curve can be associated with a geometric phase. Relations between the time evolution of a discrete curve and a geometric phase are discussed. Our constructive formalism is applied to classical and quantum Heisenberg spin chains. Keywords:
Motion of discrete curves: Berry phase; Nonlinear
sigma model; Haldane gap
1. Introduction The study of space curves finds applications in various branches of physics. As illustrative examples of a space curve we mention the propagation of light in a twisted optical fiber [ 1,2], the time evolution of the (normalized) classical spin vector field of a onedimensional Heisenberg chain and the vector field of the O(3) nonlinear sigma model in field theory [36]. This latter model is known to be related to onedimensional antiferromagnets. In recent years, there has been much interest in the notion of anholonomy in physics [ 7-91. The geometric phases and the corresponding gauge fields have attracted attention in a wide spectrum of problems in classical and quantum systems. In this Letter we investigate relations between geometric phases and moving discrete curves. Such viewpoints will be useful for a deeper understanding of the dynamics in lattice spin systems and polymer chains where the discrete nature of the systems is important. The outline of this Letter is as follows. In Section
0375-9601/96/$12.00 PI1 SO375-960
@ 1996 Eisevier Science B.V. All rights reserved
I (96)00207-l
2 we introduce a discrete curve and use its evolution to obtain an expression for the associated anholonomy density. In Section 3 we apply the motion of the discrete curve to a classical ferromagnetic chain. In Section 4, an application to one-dimensional quantum antiferromagnets is shown. In the last section we present concluding remarks.
2. Discrete curves and anholonomy We curve) integer vertex vector
consider a piecewise linear curve (a discrete in R3 [ IO,1 11. Each segment is labelled by an n which is numbered from one of the ends. A on the discrete curve is specified by the position r,,. We express r, by the recursion formula
r,+t = r, + U,, where length vector vector
(2.1)
t, is the unit “tangent” vector and h, is the of the nth segment (Fig. 1.) We define the unit b, by b, = tn+l x t,. We take n, as the unit orthogonal to t, with the direction given by
M. Hisakado, M. Wadati/ Physics Letters A 214 (1996) 252-258
Fig.
I. A discrete curve in Ii@.
n,! = b,, x t,,. We refer to n, and b, as the principal normal and binormal vectors, respectively. Then, each set of vectors, t,,, nrr and b,, forms an orthonormal frame. Neighboring frames, {t,+t, n,+r, b,+l} and {t,!, II,,, b,,}, are related by the discrete Frenet-Serret equations,
Fig. 2. Two routes (I -+ b plane (n, ~1)
253
tl and tl 4
c 4
rl in a space-time
case we write the phase as @2. The rotation angles, @I and @p2,in the two cases are @I = &,(M) + ~o(n + l,u)Au, @2 =
ro(n,u)Au
+ #+(u + AU,.
The phase difference A@=
(2.4)
A@ = @r - (Pz is given by
[ro(rr + 1,~) - ro(rl,z*)]
where S,, and T,, are rotation matrices expressed as _
4n(u + Au) - h,(u)
Au
All = j,,(u)Au,
Let u denote time and (n,u) a point in space-time. As we move from n = 110 to n = nt, a phase difference (D = Ci!_,,, c#+ develops between the natural frame and the nonrotating frame. As we move from ~0 to III along the “temporal” curve, a phase difference (D = sUy,’rc( u) du develops between the natural frame and the corresponding nonrotating frame. 70 is detined later. We consider a space-time change of the tangent to the moving discrete curve from a point a = (n, u) to a point d = (II + 1, u + AU). This may be considered as a discrete version of the work by Balakrishnan et al. [ 3 1. We choose two paths (Fig. 2). One is the path II -+ b + d, where b is the point (n + 1, u). In this case we express the phase by Qt. The other is the path N ~1 c + d. where c is the point (n, u + Au). In this
for Au -+ 0,
(2.5)
where j,,(u) = [ra(n + 1,~) - r~~(n,u) 1 - d&/d14 can be thought of as a measure of the “anholonomy density” of the system. Thus the total “anholonomy” as the system evolves in time from u = Tr to II = T: and in space from n = 1 to ~1= N is calculated as
(2.6)
In what follows, we express @ as a function of 8, and & for a moving discrete curve. We introduce
N, = (n, + ib,) exp
(2.7)
and
(2.8)
h4. Hisakado, M. Wadati/
254
Physics Letters A 214 (1996) 252-258
It turns out to be more convenient to describe the temporal evolution of the curve by t,, N, and N,’ instead of t,,, n,, and b,. We readily have N,,.N,;
N,, . t, = N; .t,, = N, . N, = 0.
= 2,
Due to (2.9) the temporal dependences lowing forms, dN,, ~ = a,,N,, + k&N: + ynt, = i&N, du
(2.9)
have the fol-
+ yntn,
dt
Fig. 3. The unit sphere of tn(u). shown.
--!! = A,N,, + p,,N; + v,,t,, du
=
+,*Nn +r,tN;).
(2.10)
Imposing the compatibility condition to (2.2) and (2.10) and choosing appropriate forms for R, and Y,,, we can obtain some integrable differential-difference equations for qn [ 10-131. Since t, - dt,,/du = 0, we can write
A trajectory
I’ and a cap I: are
in (2.14)) we can obtain another expression for j, (u), j,(u)
= --7o(n)(l
-cosB,)
- LI,sin8,.
(2.17)
Further, since tn+l = cos O,,t,,+sin $,,n,, and db,/du -CJntn -bong, we obtain
=
(t,,+~-t,)xb,=[~0(n)(l-~os~~)+U~sin8,]b,,. dt -! = \5,n,, + U,b,,. du From (2.7)-( sions,
Yn
2.11) we obtain the following
expres-
The total phase @ can therefore be written in the following forms,
=-@
= -(V, + iUn,)exp
%f = -Kt,,
(2.18)
(2.11)
+ To(n)&,,
TZ
(2.12)
h, = -c/d,, - Bonn,
N
=-I c du
J
= -1
[~(n)(l
-cosi?,)U,sin&l
k=l
T
where (2.13)
AkRk,
dugb,,.
(@n+, -tn)
x 2).
(2.19)
T1 Thus, the anholonomy density defined by (2.5) can be directly computed from (2.13). The result is
When t, satisfies the boundary tltu)
j,,(u)
= ~(n+
1,~) - ~(n,u)
d& - du
= -A,R,, (2.14)
where A,, is the difference
operator, (2.15)
A,,% = R,+I - R,. Using the time dependence W+, du=
q(n+
of &,
1) -~(n)cos8,+U,,sin8,,
(2.16)
= t(u),
tn(Tl) = tnt~z),
conditions
CN(U) s to(u), (2.20)
the total phase @ is the area of a cap X which contains to. The cap C is a region on the unit sphere whose boundary is a trajectory r parameterized by t(u) (Fig. 3). For appropriate boundary conditions, such that t,, - mo (a constant vector) at infinity in space-time (2.19), becomes the Pontryagin index. Expression (2.19) for @ indicates that @ can be interpreted as a classical analog of the quantum geomet-
255
M. Hisakado, M. Wadati/Physics Letters A 214 (1996) 252-258
ric phase introduced by Berry. Since Eq. (2.10) R,, = -iiN,S . dN,,/du, we obtain
gives
using the Hasimoto transformation [ I 51. The NLS equation is a completely integrable system. Using the discrete Frenet-Serret equations (2.2) in (3.4), we obtain
(2.21)
dtn
hu=Replacing the complex unit vector N,,/& tum state in), we see that
by a quan-
sin on_ I sin &_ I n,,
+ (sine,,
- sin8,_t
VI = -sin0,_i
sin&i,
ufl = sint), - sine,_i plays the role of a local Berry phase at the point n, as seen in the continuous case [ 3,7]. The formulation is general so far. We shall apply it to spin chains in the following two sections.
spin system whose
H= -JAW,,+,,
(3.1)
6 __ = JIS,, x (S-I + S,,,,)]. The spin vector S,, satisfies a local constraint (3.3)
Detining the unit vector l,, = S,/S and adimensionless variable II = JSt, we obtain
dt,, - =t,, x (t ,,_I +t,+,). du
On the other hand, if we set the velocity fields as V, = -2sin&
tan(+6,), (3.7)
CJn=2tan(+8,,)cos~,-2tan($tV,_i),
dq, = i[q,+i du
- 2qn + qn-i + M2(qn+t
Schrodinger
+ +-I
)I.
This equation is integrable and in the continuous limit, Eq. (3.8)) reduces to the NLS equation. Both cases (3.6) and (3.7) give, in the continuous limit, V,, = -KT, u, = aK/aS, where K, r and s are the curvature, torsion and arclength. For (3.6), sin 0,, corresponds to the curvature and for (3.7 ) .2 tan( it’?,,) corresponds to the curvature.
4. Effective action for one-dimensional antiferromagnets
dt
for all II.
(3.6)
(3.8)
where S,, denotes the classical spin vector at the nth lattice site, and J represents the nearest-neighbor exchange coupling. We assume that J > 0, which corresponds to ferromagnetic coupling. The time evolution of S,, ( 1) obtained from the Hamiltonian (3.1) is
Si = S’ = const
cos~,,_i.
we can obtain the lattice nonlinear (LNLS2) equation [ 10,13,14]
chain
We consider a one-dimensional Hamiltonian is
(3.5)
)b,.
Thus, we have (2.22)
3. Ferromagnetic
cos&i
quantum
We study quantum antiferromagnets on lattices. Consider a spin chain occupied by spin-s degrees of freedom. Each site of the lattice is labeled by an integer j = 0,. . . , N. We assume that the number of sites, N + 1, is even. The real-time action is given by [ 161
(3.4)
Note that t,, can be identified with the tangent vector to a discrete curve evolving in time. It is known [ 141 that C3.4) does not have a zero-curvature representation and therefore is not integrable. In the continuous limit we obtain the nonlinear Schriidinger (NLS) equation,
S.M[AI = s~&dzlriOl ;=O T _
N .Is*A(j,xo)
dxo s 0
c J=o
.A(j+
1.x0),
(4.1)
256
M. Hi&ado,
M. Waduti/Physics
where Swz [ n(j) ] is the Wess-Zumino action, J > 0 the nearest-neighbor antiferromagnetic coupling constant and ii the unit vector on a unit sphere S*. The Wess-Zumino action is defined as
Letters A 214 (19961252-258
Si~[n]
=s~(-l)i+'Swz[n(j)]
.i=o -
i
Js*
[n(j,xa) J
0
Swz(ii) T2
1,X0)1*.
(4.6)
Kz
-.i J du
Tl
dkir(k,u)
- [&Et(k,u)
x &fi(k,u)],
KI
(4.2) with the boundary
= fi(u),
ii(k,T,)
= ii(k,T*).
ii(Kz,u)
= fit), (4.3)
The vector fi(j, u) can be identified with the tangent t.i (u) of a discrete curve evolving in time. Because the Wess-Zumino action is the area of the sphere bounded by the trajectory of ii(j, u), we apply (2.19) to describe the Wess-Zumino action as fi
=
JC
N
du
T
C~,,+I -t,,)
We now split the staggered spin field n into a slowly varying part m(j), the order parameter field, and a small part, 2 (j), which roughly represents the average spin. Hence, we may write it as [ 171 n(j)
conditions
Ei(K,,lO
Swz(t,,)
- n(j +
.i=o
.(bn
x bw).
(4.4)
k=l
= m(j)
+
(4.7)
(-lPaol(j),
where ao denotes the lattice spacing (i.e. a short distance cutoff). The constraint n* = 1 and the requirement that the order-parameter field m should obey the same constraint, m* = 1, demand that the vectors m and 1 be orthogonal, m-1
=O.
(4.8)
Recall that Swz is essentially the area of the sphere bounded by the trajectory n( x, no) (at each point x) on the spin manifold. Thus the variation SSwz due to the small change in the trajectory Sn is simply equal to SSwz = Sn -
( $ >. n x
(4.9)
0
The Wess-Zumino
term in (4.6) can be rewritten
as
N
In this form, the boundary condition is taken to be (2.20). Terms like (4.4) are generically called WessZumino terms and are also referred to as ChernSimons terms. We suppose that the system is close to the NCel state, and therefore we change ii(j) as
ii(j)
+ (-l)j+‘n(j).
s
c
(6l)j*‘Swz[n(j)l
j=O
(N+l)/*
=s
C {Swz[n(2r+
I)1 -&z[n(2r)l}.
Go
(4.10) From (4.7) we have
(4.5) n(2r
On a bipartite lattice, transformation (4.5) changes the sign of the exchange coupling to a ferromagnetic one. The Wess-Zumino terms are odd under (4.5) and thus staggered. We see that it is the Wess-Zumino term, a purely quantum mechanical effect, which distinguishes ferromagnets from antiferromagnets. After staggering the spins we obtain, up to an additive constant,
+ 1) - n(2r)
=m(2r+
1) -m(2r)
+ao[Z(2r+
1) +1(2r)].
(4.11) Using (4.11) and neglecting (4.10), we obtain
2 j=l
(-l)j+'&z[n(j)l
higher-order
terms in
M. ffisahmio, (+I,/2
hf. Wadati/Physics
T
-$aos
N
d-x,-,6n (2r, xc)
‘v s ES i-=0
0
n(2r,xo)
x
IN-I),/’
‘f
^”
.s
dn(2r, xo) - &)st
dxo
CJ’
(do&t x &t,
(4.14)
- (?I($ x a,q,L,
where L is the length of the spin chain, and $0 and kJr denote partial differentiation with respect to _YO xrd xi. Notice that ~9; denotes ai a,. Thus. one finds
+ I) - m(2r)
0
+m[l(2r+
dx2
J
dxo(m(2r
GO
J t.
dx2t - (Jot x c#t)
= &OS
*
257
Letters A 214 (1996) 252-258
I) f1(2r)]} (4.12)
/=I
WC can identify m(2r) with the tangent t2,. of a discrete curve evolving in time. Then the Wess-Zumino term is written as
is
”
J
t.
dx”
+ ;aos
c ,/=I
T
T
CJ’
iv s
dxo(t?-r+l
IdI
-
t2r)
.
k 0
;
c
I
r-=u
d.xotk+l . (tn x aotk),
Cs
0
(N-l)/2
+.s
(4.15)
The third term of (4.15) comes from ;I difference of the discretization. We have discretized the WessZumino terms in two ways,
( ~- 1) j”&&z(j)l
(iv--l),‘?_
J
(do1 x a:t).
J s
ds’ I . (t x dot)
d2xt-(c30tx&t)+s
T dxo(t2r+l
-
t2r)
dxo(tk+, CJ
- tk) * (bk
X
dobk).
(4.16)
k 0
*[(l:rX&)-(h2,.Xf?J]
tN-~l +
j’?
dxoao[Z(2r+ xi’,-=O
.
7
.s
1) +1(2r)l
0
(I?,. x 2).
In the continuum
lim .r
(4.13)
limit we have
dXo(t2rtr
-
t2r)
The former is the simple discretization. The latter is due to the discretization in the geometric significance as seen in (4.4). In the continuous limit, the two yield the same, but in the order of an there is a difference which gives the third term of (4.15). The second term of (4.14) can be interpreted as the Berry phase of two free spins, located at the endpoints 0 and L. If we impose the boundary condition that t approaches asymptotically to a constant vector t --t to at infinity in the space-time, the second term of (4.14) can be ignored. Similarly, the continuum limit of the potential cnergy terms can be given by T
. I ( t~v
= lim s
X dot2r)
-
(
b?r x &bzr) I
dxot70(2r)(l
lim iJs’2 - cos@2,)
1
-- &h2
/ dxo(t,
I’
- t,,, )’
d2x[ (d,t)’
+ 419,
(4.17)
258
M. Hisakado, M. Wadati/Physics
Collecting (4.15) and (4.17), we find a Lagrangian density involving both the order parameter field t and the local spin-density 1. The fluctuations in the average spin-density I can be integrated out. The result is the Lagrangian density of the nonlinear sigma model,
+ $st - (do x a,t) + +ost * (dot x aft>, where g and ~1,~ are, respectively, and the spin wave velocity,
g2
(4.18)
the coupling constant
Letters A 214 (19961252-258
grability. For the quantum Heisenberg chain we have obtained a new term in the nonlinear sigma model. The term originates from different discretization schemes. Whether this term is relevant or irrelevant in the quantum spin chains and the geometric meaning of the term is an interesting future problem.
References II 12 13
A. Tomita and R.Y. Chiao, Phys. Rev. Len. 57 ( 1989) 973. F.D.M.
Haldane, Opt. Len.
Len. 64 (1989) I:,~ =
s’
2aoJs.
(4.19)
fl0.
730.
( 1993)
2107; Phys. Rev. B. 47
Phys. Rev. 3108.
14 1 F.D.M. Haldane, Phys. Lett. A. 93 ( 1983) 3359; Phys. Rev. Lett. 50 (1983)
The second term of (4.18) is the well known Pontryagin index. The last term is a new term from the cutoff
I I ( 1986)
R. Balakrishnan, A.R. Bishop and R. Dandoloff
[5]
1.Affleck,
161 F.D.M.
1153.
Phys. Rev. Lett. 54
Haldane,Phys.
( 1985)
966.
Rev. Len. 61 (1988)
17) M.V. Berry, Proc. R. Sot. A 392 (1984)
1039.
4.5.
181 Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58
( 1987)
1593. 191 J. Samuel and R. Bhandari, Phys. Rev. Len. 60 ( 1988) 2339.
5. Concluding
remarks
We have studied the time evolution of a discrete curve. We have shown that the evolution is associated with an anholonomy density j,!(u). The total phase s du c jk ( U) becomes the discretized representation of the Pontryagin index. Applying the general formalism to the classical Heisenberg chain, we have shown that the classical discrete Heisenberg chain and the lattice nonlinear Schriidinger equation have different properties in the geometrical significance and the inte-
I 10 I A. Doliwa and PM. Santini. J. Math. Phys. 36 ( 1995)
II I I
M. Hisakado, K. Nakayama
13.59.
and M. Wadati, J. Phys. Sec.
Japan 64 ( 1995) 2390.
[ I21 1I3 ] I 14)
M. Hisakado and M. Wadati, J. Phys. Sot. Japan. in press. M. Hisakado and M. Wadati, preprint. L.D. Faddeev and L.A. Takhtajan. Hamiltonian the theory of solitons (Springer,
[ 1s I
M. Lakshmanan,
116]
E. Fradkin,
theories of condensed Reading, MA,
I99
methods in
1980).
Phys. Lett. A. 64 (1977)
Field
(Addison-Wesley.
Berlin,
S3. matter
systems
I)
117 I I.K. Affleck, in: Strings, fields and critical phenomena, Les Houches Summer School 1988. Session XLIX, and J. Zin-Justin (Springer.
Berlin,
1990).
eds. E. Brezin
PHYSICS
ELSEVIER
LETTERS
A
Physics Letters A 2 14 ( 1996) 252-258
Moving discrete curve and geometric phase Masato Hisakado, Miki Wadati Depurtment
of Physics, Faculty
of Science, University
ofTokyo,
7-3-I
Hongo,
1 November 1995; revised manuscript
Received
received 25 January 1996; accepted Communicated by A.R. Bishop
Bunkyo-ku,
Tokyo 113. Japun
for publication
26 February
1996
Abstract The study of space curves finds many applications ih physics, such as optical fibers, magnetic spin chains, and vortex filaments in a fluid. The time evolution of a space curve can be associated with a geometric phase. Relations between the time evolution of a discrete curve and a geometric phase are discussed. Our constructive formalism is applied to classical and quantum Heisenberg spin chains. Keywords:
Motion of discrete curves: Berry phase; Nonlinear
sigma model; Haldane gap
1. Introduction The study of space curves finds applications in various branches of physics. As illustrative examples of a space curve we mention the propagation of light in a twisted optical fiber [ 1,2], the time evolution of the (normalized) classical spin vector field of a onedimensional Heisenberg chain and the vector field of the O(3) nonlinear sigma model in field theory [36]. This latter model is known to be related to onedimensional antiferromagnets. In recent years, there has been much interest in the notion of anholonomy in physics [ 7-91. The geometric phases and the corresponding gauge fields have attracted attention in a wide spectrum of problems in classical and quantum systems. In this Letter we investigate relations between geometric phases and moving discrete curves. Such viewpoints will be useful for a deeper understanding of the dynamics in lattice spin systems and polymer chains where the discrete nature of the systems is important. The outline of this Letter is as follows. In Section
0375-9601/96/$12.00 PI1 SO375-960
@ 1996 Eisevier Science B.V. All rights reserved
I (96)00207-l
2 we introduce a discrete curve and use its evolution to obtain an expression for the associated anholonomy density. In Section 3 we apply the motion of the discrete curve to a classical ferromagnetic chain. In Section 4, an application to one-dimensional quantum antiferromagnets is shown. In the last section we present concluding remarks.
2. Discrete curves and anholonomy We curve) integer vertex vector
consider a piecewise linear curve (a discrete in R3 [ IO,1 11. Each segment is labelled by an n which is numbered from one of the ends. A on the discrete curve is specified by the position r,,. We express r, by the recursion formula
r,+t = r, + U,, where length vector vector
(2.1)
t, is the unit “tangent” vector and h, is the of the nth segment (Fig. 1.) We define the unit b, by b, = tn+l x t,. We take n, as the unit orthogonal to t, with the direction given by
M. Hisakado, M. Wadati/ Physics Letters A 214 (1996) 252-258
Fig.
I. A discrete curve in Ii@.
n,! = b,, x t,,. We refer to n, and b, as the principal normal and binormal vectors, respectively. Then, each set of vectors, t,,, nrr and b,, forms an orthonormal frame. Neighboring frames, {t,+t, n,+r, b,+l} and {t,!, II,,, b,,}, are related by the discrete Frenet-Serret equations,
Fig. 2. Two routes (I -+ b plane (n, ~1)
253
tl and tl 4
c 4
rl in a space-time
case we write the phase as @2. The rotation angles, @I and @p2,in the two cases are @I = &,(M) + ~o(n + l,u)Au, @2 =
ro(n,u)Au
+ #+(u + AU,.
The phase difference A@=
(2.4)
A@ = @r - (Pz is given by
[ro(rr + 1,~) - ro(rl,z*)]
where S,, and T,, are rotation matrices expressed as _
4n(u + Au) - h,(u)
Au
All = j,,(u)Au,
Let u denote time and (n,u) a point in space-time. As we move from n = 110 to n = nt, a phase difference (D = Ci!_,,, c#+ develops between the natural frame and the nonrotating frame. As we move from ~0 to III along the “temporal” curve, a phase difference (D = sUy,’rc( u) du develops between the natural frame and the corresponding nonrotating frame. 70 is detined later. We consider a space-time change of the tangent to the moving discrete curve from a point a = (n, u) to a point d = (II + 1, u + AU). This may be considered as a discrete version of the work by Balakrishnan et al. [ 3 1. We choose two paths (Fig. 2). One is the path II -+ b + d, where b is the point (n + 1, u). In this case we express the phase by Qt. The other is the path N ~1 c + d. where c is the point (n, u + Au). In this
for Au -+ 0,
(2.5)
where j,,(u) = [ra(n + 1,~) - r~~(n,u) 1 - d&/d14 can be thought of as a measure of the “anholonomy density” of the system. Thus the total “anholonomy” as the system evolves in time from u = Tr to II = T: and in space from n = 1 to ~1= N is calculated as
(2.6)
In what follows, we express @ as a function of 8, and & for a moving discrete curve. We introduce
N, = (n, + ib,) exp
(2.7)
and
(2.8)
h4. Hisakado, M. Wadati/
254
Physics Letters A 214 (1996) 252-258
It turns out to be more convenient to describe the temporal evolution of the curve by t,, N, and N,’ instead of t,,, n,, and b,. We readily have N,,.N,;
N,, . t, = N; .t,, = N, . N, = 0.
= 2,
Due to (2.9) the temporal dependences lowing forms, dN,, ~ = a,,N,, + k&N: + ynt, = i&N, du
(2.9)
have the fol-
+ yntn,
dt
Fig. 3. The unit sphere of tn(u). shown.
--!! = A,N,, + p,,N; + v,,t,, du
=
+,*Nn +r,tN;).
(2.10)
Imposing the compatibility condition to (2.2) and (2.10) and choosing appropriate forms for R, and Y,,, we can obtain some integrable differential-difference equations for qn [ 10-131. Since t, - dt,,/du = 0, we can write
A trajectory
I’ and a cap I: are
in (2.14)) we can obtain another expression for j, (u), j,(u)
= --7o(n)(l
-cosB,)
- LI,sin8,.
(2.17)
Further, since tn+l = cos O,,t,,+sin $,,n,, and db,/du -CJntn -bong, we obtain
=
(t,,+~-t,)xb,=[~0(n)(l-~os~~)+U~sin8,]b,,. dt -! = \5,n,, + U,b,,. du From (2.7)-( sions,
Yn
2.11) we obtain the following
expres-
The total phase @ can therefore be written in the following forms,
=-@
= -(V, + iUn,)exp
%f = -Kt,,
(2.18)
(2.11)
+ To(n)&,,
TZ
(2.12)
h, = -c/d,, - Bonn,
N
=-I c du
J
= -1
[~(n)(l
-cosi?,)U,sin&l
k=l
T
where (2.13)
AkRk,
dugb,,.
(@n+, -tn)
x 2).
(2.19)
T1 Thus, the anholonomy density defined by (2.5) can be directly computed from (2.13). The result is
When t, satisfies the boundary tltu)
j,,(u)
= ~(n+
1,~) - ~(n,u)
d& - du
= -A,R,, (2.14)
where A,, is the difference
operator, (2.15)
A,,% = R,+I - R,. Using the time dependence W+, du=
q(n+
of &,
1) -~(n)cos8,+U,,sin8,,
(2.16)
= t(u),
tn(Tl) = tnt~z),
conditions
CN(U) s to(u), (2.20)
the total phase @ is the area of a cap X which contains to. The cap C is a region on the unit sphere whose boundary is a trajectory r parameterized by t(u) (Fig. 3). For appropriate boundary conditions, such that t,, - mo (a constant vector) at infinity in space-time (2.19), becomes the Pontryagin index. Expression (2.19) for @ indicates that @ can be interpreted as a classical analog of the quantum geomet-
255
M. Hisakado, M. Wadati/Physics Letters A 214 (1996) 252-258
ric phase introduced by Berry. Since Eq. (2.10) R,, = -iiN,S . dN,,/du, we obtain
gives
using the Hasimoto transformation [ I 51. The NLS equation is a completely integrable system. Using the discrete Frenet-Serret equations (2.2) in (3.4), we obtain
(2.21)
dtn
hu=Replacing the complex unit vector N,,/& tum state in), we see that
by a quan-
sin on_ I sin &_ I n,,
+ (sine,,
- sin8,_t
VI = -sin0,_i
sin&i,
ufl = sint), - sine,_i plays the role of a local Berry phase at the point n, as seen in the continuous case [ 3,7]. The formulation is general so far. We shall apply it to spin chains in the following two sections.
spin system whose
H= -JAW,,+,,
(3.1)
6 __ = JIS,, x (S-I + S,,,,)]. The spin vector S,, satisfies a local constraint (3.3)
Detining the unit vector l,, = S,/S and adimensionless variable II = JSt, we obtain
dt,, - =t,, x (t ,,_I +t,+,). du
On the other hand, if we set the velocity fields as V, = -2sin&
tan(+6,), (3.7)
CJn=2tan(+8,,)cos~,-2tan($tV,_i),
dq, = i[q,+i du
- 2qn + qn-i + M2(qn+t
Schrodinger
+ +-I
)I.
This equation is integrable and in the continuous limit, Eq. (3.8)) reduces to the NLS equation. Both cases (3.6) and (3.7) give, in the continuous limit, V,, = -KT, u, = aK/aS, where K, r and s are the curvature, torsion and arclength. For (3.6), sin 0,, corresponds to the curvature and for (3.7 ) .2 tan( it’?,,) corresponds to the curvature.
4. Effective action for one-dimensional antiferromagnets
dt
for all II.
(3.6)
(3.8)
where S,, denotes the classical spin vector at the nth lattice site, and J represents the nearest-neighbor exchange coupling. We assume that J > 0, which corresponds to ferromagnetic coupling. The time evolution of S,, ( 1) obtained from the Hamiltonian (3.1) is
Si = S’ = const
cos~,,_i.
we can obtain the lattice nonlinear (LNLS2) equation [ 10,13,14]
chain
We consider a one-dimensional Hamiltonian is
(3.5)
)b,.
Thus, we have (2.22)
3. Ferromagnetic
cos&i
quantum
We study quantum antiferromagnets on lattices. Consider a spin chain occupied by spin-s degrees of freedom. Each site of the lattice is labeled by an integer j = 0,. . . , N. We assume that the number of sites, N + 1, is even. The real-time action is given by [ 161
(3.4)
Note that t,, can be identified with the tangent vector to a discrete curve evolving in time. It is known [ 141 that C3.4) does not have a zero-curvature representation and therefore is not integrable. In the continuous limit we obtain the nonlinear Schriidinger (NLS) equation,
S.M[AI = s~&dzlriOl ;=O T _
N .Is*A(j,xo)
dxo s 0
c J=o
.A(j+
1.x0),
(4.1)
256
M. Hi&ado,
M. Waduti/Physics
where Swz [ n(j) ] is the Wess-Zumino action, J > 0 the nearest-neighbor antiferromagnetic coupling constant and ii the unit vector on a unit sphere S*. The Wess-Zumino action is defined as
Letters A 214 (19961252-258
Si~[n]
=s~(-l)i+'Swz[n(j)]
.i=o -
i
Js*
[n(j,xa) J
0
Swz(ii) T2
1,X0)1*.
(4.6)
Kz
-.i J du
Tl
dkir(k,u)
- [&Et(k,u)
x &fi(k,u)],
KI
(4.2) with the boundary
= fi(u),
ii(k,T,)
= ii(k,T*).
ii(Kz,u)
= fit), (4.3)
The vector fi(j, u) can be identified with the tangent t.i (u) of a discrete curve evolving in time. Because the Wess-Zumino action is the area of the sphere bounded by the trajectory of ii(j, u), we apply (2.19) to describe the Wess-Zumino action as fi
=
JC
N
du
T
C~,,+I -t,,)
We now split the staggered spin field n into a slowly varying part m(j), the order parameter field, and a small part, 2 (j), which roughly represents the average spin. Hence, we may write it as [ 171 n(j)
conditions
Ei(K,,lO
Swz(t,,)
- n(j +
.i=o
.(bn
x bw).
(4.4)
k=l
= m(j)
+
(4.7)
(-lPaol(j),
where ao denotes the lattice spacing (i.e. a short distance cutoff). The constraint n* = 1 and the requirement that the order-parameter field m should obey the same constraint, m* = 1, demand that the vectors m and 1 be orthogonal, m-1
=O.
(4.8)
Recall that Swz is essentially the area of the sphere bounded by the trajectory n( x, no) (at each point x) on the spin manifold. Thus the variation SSwz due to the small change in the trajectory Sn is simply equal to SSwz = Sn -
( $ >. n x
(4.9)
0
The Wess-Zumino
term in (4.6) can be rewritten
as
N
In this form, the boundary condition is taken to be (2.20). Terms like (4.4) are generically called WessZumino terms and are also referred to as ChernSimons terms. We suppose that the system is close to the NCel state, and therefore we change ii(j) as
ii(j)
+ (-l)j+‘n(j).
s
c
(6l)j*‘Swz[n(j)l
j=O
(N+l)/*
=s
C {Swz[n(2r+
I)1 -&z[n(2r)l}.
Go
(4.10) From (4.7) we have
(4.5) n(2r
On a bipartite lattice, transformation (4.5) changes the sign of the exchange coupling to a ferromagnetic one. The Wess-Zumino terms are odd under (4.5) and thus staggered. We see that it is the Wess-Zumino term, a purely quantum mechanical effect, which distinguishes ferromagnets from antiferromagnets. After staggering the spins we obtain, up to an additive constant,
+ 1) - n(2r)
=m(2r+
1) -m(2r)
+ao[Z(2r+
1) +1(2r)].
(4.11) Using (4.11) and neglecting (4.10), we obtain
2 j=l
(-l)j+'&z[n(j)l
higher-order
terms in
M. ffisahmio, (+I,/2
hf. Wadati/Physics
T
-$aos
N
d-x,-,6n (2r, xc)
‘v s ES i-=0
0
n(2r,xo)
x
IN-I),/’
‘f
^”
.s
dn(2r, xo) - &)st
dxo
CJ’
(do&t x &t,
(4.14)
- (?I($ x a,q,L,
where L is the length of the spin chain, and $0 and kJr denote partial differentiation with respect to _YO xrd xi. Notice that ~9; denotes ai a,. Thus. one finds
+ I) - m(2r)
0
+m[l(2r+
dx2
J
dxo(m(2r
GO
J t.
dx2t - (Jot x c#t)
= &OS
*
257
Letters A 214 (1996) 252-258
I) f1(2r)]} (4.12)
/=I
WC can identify m(2r) with the tangent t2,. of a discrete curve evolving in time. Then the Wess-Zumino term is written as
is
”
J
t.
dx”
+ ;aos
c ,/=I
T
T
CJ’
iv s
dxo(t?-r+l
IdI
-
t2r)
.
k 0
;
c
I
r-=u
d.xotk+l . (tn x aotk),
Cs
0
(N-l)/2
+.s
(4.15)
The third term of (4.15) comes from ;I difference of the discretization. We have discretized the WessZumino terms in two ways,
( ~- 1) j”&&z(j)l
(iv--l),‘?_
J
(do1 x a:t).
J s
ds’ I . (t x dot)
d2xt-(c30tx&t)+s
T dxo(t2r+l
-
t2r)
dxo(tk+, CJ
- tk) * (bk
X
dobk).
(4.16)
k 0
*[(l:rX&)-(h2,.Xf?J]
tN-~l +
j’?
dxoao[Z(2r+ xi’,-=O
.
7
.s
1) +1(2r)l
0
(I?,. x 2).
In the continuum
lim .r
(4.13)
limit we have
dXo(t2rtr
-
t2r)
The former is the simple discretization. The latter is due to the discretization in the geometric significance as seen in (4.4). In the continuous limit, the two yield the same, but in the order of an there is a difference which gives the third term of (4.15). The second term of (4.14) can be interpreted as the Berry phase of two free spins, located at the endpoints 0 and L. If we impose the boundary condition that t approaches asymptotically to a constant vector t --t to at infinity in the space-time, the second term of (4.14) can be ignored. Similarly, the continuum limit of the potential cnergy terms can be given by T
. I ( t~v
= lim s
X dot2r)
-
(
b?r x &bzr) I
dxot70(2r)(l
lim iJs’2 - cos@2,)
1
-- &h2
/ dxo(t,
I’
- t,,, )’
d2x[ (d,t)’
+ 419,
(4.17)
258
M. Hisakado, M. Wadati/Physics
Collecting (4.15) and (4.17), we find a Lagrangian density involving both the order parameter field t and the local spin-density 1. The fluctuations in the average spin-density I can be integrated out. The result is the Lagrangian density of the nonlinear sigma model,
+ $st - (do x a,t) + +ost * (dot x aft>, where g and ~1,~ are, respectively, and the spin wave velocity,
g2
(4.18)
the coupling constant
Letters A 214 (19961252-258
grability. For the quantum Heisenberg chain we have obtained a new term in the nonlinear sigma model. The term originates from different discretization schemes. Whether this term is relevant or irrelevant in the quantum spin chains and the geometric meaning of the term is an interesting future problem.
References II 12 13
A. Tomita and R.Y. Chiao, Phys. Rev. Len. 57 ( 1989) 973. F.D.M.
Haldane, Opt. Len.
Len. 64 (1989) I:,~ =
s’
2aoJs.
(4.19)
fl0.
730.
( 1993)
2107; Phys. Rev. B. 47
Phys. Rev. 3108.
14 1 F.D.M. Haldane, Phys. Lett. A. 93 ( 1983) 3359; Phys. Rev. Lett. 50 (1983)
The second term of (4.18) is the well known Pontryagin index. The last term is a new term from the cutoff
I I ( 1986)
R. Balakrishnan, A.R. Bishop and R. Dandoloff
[5]
1.Affleck,
161 F.D.M.
1153.
Phys. Rev. Lett. 54
Haldane,Phys.
( 1985)
966.
Rev. Len. 61 (1988)
17) M.V. Berry, Proc. R. Sot. A 392 (1984)
1039.
4.5.
181 Y. Aharonov and J. Anandan, Phys. Rev. Lett. 58
( 1987)
1593. 191 J. Samuel and R. Bhandari, Phys. Rev. Len. 60 ( 1988) 2339.
5. Concluding
remarks
We have studied the time evolution of a discrete curve. We have shown that the evolution is associated with an anholonomy density j,!(u). The total phase s du c jk ( U) becomes the discretized representation of the Pontryagin index. Applying the general formalism to the classical Heisenberg chain, we have shown that the classical discrete Heisenberg chain and the lattice nonlinear Schriidinger equation have different properties in the geometrical significance and the inte-
I 10 I A. Doliwa and PM. Santini. J. Math. Phys. 36 ( 1995)
II I I
M. Hisakado, K. Nakayama
13.59.
and M. Wadati, J. Phys. Sec.
Japan 64 ( 1995) 2390.
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M. Hisakado and M. Wadati, J. Phys. Sot. Japan. in press. M. Hisakado and M. Wadati, preprint. L.D. Faddeev and L.A. Takhtajan. Hamiltonian the theory of solitons (Springer,
[ 1s I
M. Lakshmanan,
116]
E. Fradkin,
theories of condensed Reading, MA,
I99
methods in
1980).
Phys. Lett. A. 64 (1977)
Field
(Addison-Wesley.
Berlin,
S3. matter
systems
I)
117 I I.K. Affleck, in: Strings, fields and critical phenomena, Les Houches Summer School 1988. Session XLIX, and J. Zin-Justin (Springer.
Berlin,
1990).
eds. E. Brezin