Download Semiclassical approximation of excitations in spin-1 Heisenberg antiferromagnets

Semiclassical treatment of quasiparticle excitations in spin S = 1
antiferromagnetic chain
Girish Sharma† , Shreyas Patankar∗
†
∗
Satyasai University, Bangalore;
Chennai Mathematical Institute, Chennai
Report of project completed under Prof. Kedar Damle, TIFR
Abstract
We briefly discuss semiclassical dynamics of quasi-particles in a spin S = 1 antiferromagnetic chain. It is shown that
the relaxation function R(x, t) as evaluated analytically in [2] and [3] also holds for our case. This result can be further
used to calculate the dynamical structure factor in the semiclassical approximation exactly. The exact evaluation of
dynamical structure factor, S(k, w) is expected to be more reliable than the approximate semiclassical result for this
quantity in [1]. It is also of interest to compare this exact semiclassical evaluation with the integrable quantum field
theory calculation done by Essler[4].
Contents
1 Introduction
2 Path integral and coherent states
2.1 Feynman path integral . . . . . .
2.2 Coherent state representation . .
4
3.1 Interaction term . . . . . . . . .
3.2 Berry Phase . . . . . . . . . . . .
1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
2
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
4 Dispersion relation for quasiparticles
6
5 Thermal correlation
5.1 Counting collisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
7
6 Analytic expression for R(x,t)
8
7 Correlation in sine-Gordon model
9
8 Dynamical susceptibility and structure factor
10
8.1 Qualitative form and symmetry of structure factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
9 Conclusion
1
11
Introduction
Inelastic neutron scattering is used as a tool for observation in condensed matter physics. It offers an advantage over
other scattering probes such as X-rays, as neutrons (having intrinsic magnetic dipole moment, unlike photons) are able
to interact with the magnetic lattice excitations. The scattering cross-section for these interactions is proportional to the
dynamical structure factor S(q, ω), which can be shown to be the Fourier transform of the two-point lattice correlation
function C(x, t).[5] We wish to obtain an expression for the structure factor for a given system.
The Hamiltonian for an anti-ferromagnetic system in one dimension with nearest-neighbour interaction is given by
X
~i · S
~i+1
H=J
S
i
1
(1)
with J > 0. Assuming a long-distance effective solution, we shall show that this Hamltonian can be reduced to that of
the quantum rotor model. Now, the rotor model Hamiltonian can be treated as a perturbation on a ground state of l = 0
chain, with the perturbation written in terms of suitable creation and annihilation operators. Thus, we can call l = 1
degenerate excitations as quasi particles, with excitation gap ∆.
At low temperatures, (T ∆), the density of excitations will be low. Following ref[1], we assume that these quasi-particles
can be treated as classical colliding particles, with the S - matrix for the collision being identically −1. Thus, the problem
of calculating the two point correlation function, is reduced to the problem of counting the expectation value of the matrix
element of collisions and the total number of such collisions. We also compare this problem with the similar problems of
soliton collisions in the sine-Gordon model[3] and domain collisions in the quantum Potts model.[2]
Essler and Konik(2008) obtained an expression for the dynamical susceptibility from exact calculation using integrable
field model. We qualitatively compare the results of our semi-classical approximation with the exact result of ref[4].
2
2.1
Path integral and coherent states
Feynman path integral
Path integral representation
is an elegant way of re-writing Z = Tr e−βH . Consider 1-D system of spins with the above
X
~i · S
~j
S
Hamiltonian H = J
hiji
Now suppose we write Z = Tr e−βH = Tr e|−H .{z
. . e−H} where M = β. Inserting complete basis set {α},
M
Z=
−1
X MY
αk+1 e−H αk
(2)
α0 k=0
such that |αM i = |α0 i
Let us use the notation αk+1 e−H αk = e−S[αk ] . Further, take discrete system of vectors αk to a continuous one α(τ )
with α(0) = α(β). This can be visualized as a collection of arbitrary “paths” in α-space,
Note that we can write X
Z
S[αk ] =
dτ S[α(τ )]. We can write “sum over all paths” as a path integral
k
Z
Z=
Dα(τ ) e−
Rβ
0
S[α(τ )]dτ
(3)
The parameter τR is also called “imaginary time”. The path integral is a sum over all possible paths in α-space, weighted
β
by the factor e− 0 S[α(τ )]dτ .
2.2
Coherent state representation
Now let us consider the particular Hamiltonian H = J
X
~i · S
~j (throughout we shall be using units with ~ = 1, kB = 1).
S
hiji
~ To write this basis in terms of
Instead of using the Sz eigenbasis {| ↑i, | ↓i}, let us consider the basis of the unit vector N
~ is a unit vector perpendicular to the projection of N
~ on the XY -plane Then we can get N
~ by
the z-basis, note that if M
~ by an angle θ Hence,
rotating ẑ about M
~ ·S
~
~ i = e−iθM
|N
| ↑i
(4)
2
~ 0 |N
~i =
It can be shown that hN
6 0 in general as the basis is overcomplete, I =
Z
~
dN
~ ihN
~ |, and that hN
~ |S|
~N
~ i = SN
~
|N
2π
1
Let us evaluate this for the special case s = . Recall that the Pauli spin matrices are given as
2
0 1
0 −i
1 0
σx =
, σy =
, σz =
1 0
i 0
0 −1
Then we can write
~ ~
~ ·S
~−
e−iθM ·S = I − (iθ)M
(iθ)2 ~ ~ 2
(M · S) + . . .
2!
Z
(5)
Z
~ (τ ). To do this, we first need to evaluate
We now wish to write the path integral in terms of the new basis, Dα → DN
E
D
~ ~
~ is a smoothly varying function of S.)
~
~ (τ + ) e−H(S)
the matrix element N
N (τ ) (assuming that H(S)
~
~ (τ + )i = |N
~ (τ )i + d |N
~ (τ )i + O(2 ) and e−H(S)
~ + O(2 ) which gives
We can write a Taylor expansion |N
= 1 − H(S)
dτ
E
D
d ~
~ ~
~ (τ )i − H(S N
~ (τ )) + O(2 ) ....(using Cayley-Hamilton theorem). Thus,
~ (τ + ) e−H(S)
hN (τ )| |N
N
N (τ ) = 1 + dτ
we can write partition function
Z
R
R
d
|N~ (τ )i+ 0β H(S N~ (τ ))]
~ (τ )e−[ 0β dτ hN~ (τ )| dτ
(6)
Z=
DN
~ (0)=N
~ (β)
N
β
~ (τ ) satisfies S ∗ = −SB ....(integrating by parts), and hence,
~ (τ ) d N
N
B
dτ 0
1
it is purely imaginary. SB is called the Berry phase. It can be similarly shown that the second term in the exponent is
purely real.
~ (τ )·S
~ d −iθ(τ )M
~ ·S
~
~ (τ )·S
~ ~ (τ ) d N
~ has the form |N
~ i = e−iθM
~ (τ ) = ↑ eiθ(τ )M
Recall that N
| ↑i. Then, N
e
dτ ↑ . We shall
dτ
use the identity for any operator O(τ ),
Z
The first term in the exponent SB =
dτ
d O(τ )
e
=
dτ
Z
1
du e(1−u)O(τ )
0
d
O(τ ) euO(τ )
dτ
(7)
~ as a function of two variables, N
~ (u, τ ), such that N
~ (0, τ ) = ẑ and N
~ (1, τ ) = N
~ (τ ). Then, we have
Now, we can write N
~
~
~ =N
~ × dN ∂ N and we can write
θM
!
!
Z 1 du Z∂uβ
Z 1 Z β
~
~
~
~
∂N
∂N
∂N
∂N
~
~
SB = iS
·
= iS
×
du
du
dτ N ×
dτ N ·
∂u
∂τ
∂u
∂τ
0
0
0
0
!
~
~
~ · ∂ N × ∂ N is the oriented area of a small square on the sphere,
This can be interpreted geometrically. Observe that N
∂u
∂τ
and hence, we can see that the double integral will give the oriented area of the part of the sphere enclosed by the loop
~ (τ ) for 0 < τ < β.
N
1
Also called the geometric phase or the Pancharatnam-Berry phase
3
Long distance effective model2
3
3.1
Interaction term
Consider an approximate solution to the Hamiltonian, where locally the spins are anti-aligned, however, over long (∼
5×lattice spacing) distances, the (anti-)alignment vector changes direction. As spins are almost anti-aligned, the total
~ to be angular momentum
angular momentum in any neighbourhood is expected to be a small quantity. Suppose we use L
~ is expected to be small compared
density, then the value of the total angular momentum in a small neighbourhood ad L
d 2
a ~ 1
to the scales of the system, that is L
S Now let ~n be the anti-alignment vector averaged over several lattice sites. By construction, ~n should be a slowly-varying
function.
~ can be split as:
Thus, by imposing normalisation, the spin direction N
r
d
2d
j
~ (xj , τ ) = (−1) ~n(xj , τ ) 1 − a L
~ j, τ)
~ 2 + a L(x
N
(8)
2
S
S
We expect that most of the contribution towards the path integral (6) should come from terms of the form (8).
We shall now subsitute this into the expression for the partition function. For the interaction term
a2d
d X
X
a2d
2
2
ja
2
~
~
~
~
L + Lj+1 − (−1)
Lj · ~nj+1 − ~nj · Lj+1 + 2 Lj · Lj+1
Ŝj · Ŝj+1 = −JS
−~nj · ~nj+1 +
J
2S 2 j
S
S
j
j
2
a2 ~nj − ~nj+1
a2
2
Now, note that −~nj · ~nj+1 =
(∂r ~n) − 1 ....(shifting from discrete to continuous.)
−1≈
2
a
2
~ j · ~nj+1 − ~nj · L
~ j+1 ≈ ~nj · L
~j − L
~ j+1 ≈ −a~n · ∂r L
~ and as this is to be multiplied by
Further, as ~n is slowly varying, L
(−1)j , it contributes only in higher order derivatives.
~ ~n are slowly varying and that L
~ is small, we get
Thus, assuming L,


Z
1
 2 2−d
2
d ~ 2
a } (∂r ~n) + 2Ja
H = −1 +
dd x JS
| {z
| {z L}
2
ρs
(9)
χ−1
⊥
where the term ρs is called the bare stiffness and χ⊥ is called the perpendicular susceptibility. Note that if we apply an
~ perpendicular to the ordering, then it couples directly to L
~ as H
~ · L.
~ Minimizing the total
external magnetic field H
~
Hamiltonian gives us L = χ⊥ H.
3.2
Berry Phase
Let us now use this approximation to expand the Berry phase term. SB = iS
X
~ j (τ )]. Recall that we can write the
A[N
j
Berry phase explicitly as SB = iS
XZ
j
0
β
Z
dτ
0
1
~ ·
du N
~
~
∂N
∂N
×
∂τ
∂u
!
. We can now use
r
a2d ~ 2 ad ~
~
~ (τ ).
N (xj , τ, u) = λj n̂(xj τ, u) 1 − 2 L
+ L(xj , τ, u), the parameter u describing the rotation from ẑ to position N
S
S
~ as A and B.
Let us call the coefficients of n̂andL
!
~
~
~
~
∂N
∂
N
∂
N
∂N
~·
and
are both perpendicular to ~n and hence L
×
=0
Now,
∂τ
∂u
∂u
∂τ
We now further !
simplify the Berry Phase term. Expanding the integral in () we obtain:
~
~
∂
N
∂
N
~ ·
N
×
=
∂u
∂τ
"
!
!#
~
~
∂
L
∂~
n
∂
L
∂~
n
~ j , τ, u) ·
+ (B)
× (A)
+ (B)
An̂(xj τ, u) + B L(x
(A)
∂u
∂u
∂τ
∂u
2
Summary of discussion presented in ref[6]
4
We further simplify the above, using the approximations we have stated earlier.
Z
Z 1 ∂
ad X β
∂~n ~
∂
∂~n
~
dτ
du
We simplify the following term: SB =
iS
~n ·
×L +
~n · L ×
S
∂τ
∂u
∂u
∂τ
0
0
j
u=1
Z
X β ∂
n̂
~×
dτ n̂ · L
The first term vanishes due to periodic boundary conditions. The second term is:iS
∂τ
0
i
u=0
~ since all the frames are being rotated up to North Pole this is actually a
At u = 0 though the term is first order in L,
much higher order term. Hence we obtain the following expression for Z:
!
Z
Z
Z β Z 1
2
1
L
∂
n̂
0
~ L
~ · n̂) exp SB
~ · (n̂ ×
Z = Dn̂ DLδ(
(n̂) −
dτ
dd x ρs [∇n̂]2 +
+ 2iL
)
2 0
2χ⊥
∂τ
0
0
~ · n̂) as an integral and perform the integral
SB
is the first term in the expansion of the Berry Phase. We can rewrite δ(L
~
over L converting the integral to standard Gaussian form. We finally obtain:
Z
Z=
Dn̂ exp
0
SB
(n̂)
1
−
2
Z
β
Z
dτ
0
0
1
!
1
2
2
d x ρs [∇n̂] + 2 (∂τ n̂)
c
d
(10)
0
The sum involving SB
is an alternating sum. We assume periodic boundary condition with N + 1 coincidng with 1. We
have to evaluate the integral of the following series:
X
∂ n̂2i+1
∂ n̂2i+1
∂ n̂2i
∂ n̂2i
n̂2i+1 ·
×
− n̂2i ·
×
∂u
∂τ
∂u
∂τ
i
0
In the continuum limit one obtains the following expression for SB
:
Z
Z β
iS L
∂ n̂ ∂ n̂
0
SB
=
dx
dτ n̂ ·
×
(11)
2 0
∂x
∂τ
0
∂ n̂ ∂ n̂
×
is the number (n) of times n̂(x, τ ) covers the sphere as x goes from 0 to L and τ goes from 0 to β.
Now n̂ ·
∂x
∂τ
Hence the integral gives us: 4πn. Hence the Berry phase turns out to be exp(iS2π).Thus the Berry Phase vanishes for
the spin one systems. A more rigorous analysis also shows the same thing. We obtain the following expression for Zef f :
Z
Z
Z
ρs
1
Zef f = D~nδ(~n2 − 1) exp − dd x dτ
(12)
(∇~n)2 + 2 (∂τ ~n)2 exp(iSB )
2
c
We have thus reduced the quantum problem in d dimensions to a classical problem in d + 1 dimension with a Berry Phase
term. With some rescaling of variables one can write the following expression:
Z
Z β
1
S=
dd x
dτ c2 [∇~n]2 + [∂τ ~n]2
(13)
2cg
0
where S is the factor inside the exponential in the expression for Z. Each of the two terms os S can be compared to the
potential and kinetic energy terms. We thus can say that the effective Hamiltonian that results in this expression of Zef f
is:
X gL2
J
i
2
+ (~ni+1 − ~ni )
H=
2
2
i
The above model is also known as the Rotor Model. At g = 0 we get a perfectly ordered state with anti-ferromagnetic
coupling. Non zero g destroys ordering. It can be proved that even for small values of g the effective contribution to the
Hamiltonian due to the second term is very small and thus can be treated as a perturbation.
The ground state of the Hamiltonian is characterized by l = 0 at each site. The first excited state is l = 1 at a single site.
This can be treated as a particle which hops around the lattice from one site to the other. Hence we obtain a N − f old
degenerate first excited state. We now include the perturbation term. Thus we need to employ degenarate perturbation
theory to lift the degeneracy. Hence we evaluate the matrix elements of −J~ni · ~ni+1 in the angular momentum basis. We
do this by expanding n̂ in SPC. The matrix elements are expected to be same between any 2 adjacent pairs and are non
zero only if |n − n0 | = 1, where n and n0 are the row and column index. Tha magnitude is found to be −J/3.
It can be proved more rigorously using field theory that these spin one antiferromagnetic chains are gapped, and we denote
the energy gap by ∆. We can now obtain the dispersion relation for these quasiparticles in the first excited state.
5
4
Dispersion relation for quasiparticles
We represent the state at each site as a ket : |pi, where p = 1, 2, .....N ; and N + 1 coincides with 1 (cyclic chain). We now
construct |ki as the following:
X
|ki = C
eikap |pi
(14)
p
where a is the lattice spacing.
1
Next we normalize |ki and obtain C = √ by using delta function.
N
X
The new basis |ki diagonlaizes V (the perturbation matrix i.e. −J
~nj · ~nj+1 ).
j
We now evaluate hk|V |ki.
hk|J
X
~nj · ~nj+1 |ki =
j
1 X
exp(ika(p − p0 ))hp0 |~nj · ~nj+1 |pi
N 0
p,p ,j
The only nonvanishing terms are |p − p0 | = ±1.
=
1 X ika
e hp|~nj · ~nj+1 |p + 1i + e−ika hp + 1|~nj · ~nj+1 |pi
N p,j
Only the sites j and j + 1 will contribute.
=
1 X ika
e hj|~nj · ~nj+1 |j + 1i + e−ika hj + 1|~nj · ~nj+1 |ji
N j
We know the matrix elements hj|~nj · ~nj+1 |j + 1i = 1/3. Hence we obtain the diagonal elements of the V matrix as:
=
2J
J X2
cos(ka) =
cos(ka)
N j 3
3
Which gives
H=−
2J
cos(ka)
3
(15)
It can be shown that V is diagonal in this basis by evaluating hk 0 |V |ki which vanishes unless k = k 0 .
k 2 a2
, and hence we can write dispersion relation as
For small momenta, we can write cos(ka) ≈ 1 −
2
ε(p) = ∆ +
5
p2
+ O(p3 )
2mef f
(16)
Thermal correlation
Observe that the matrix elements hl = 0|nz |l = 1, m = 0i, hl = 0|nx |l = 1, m = ±1i, hl = 0|ny |l = 1, m = ±1i are
nonvanishing, we can write the operator n̂ as n0 = nz , n+ = nx − iny and n− = nx + iny such that n0 is the sum of
creation and annihilation operators for excited l = 1 states with m = 0, similarly, n+ is sum of annihilation operator for
m = 1 and creation operator for m = −1 and n+ is sum of creation operator for m = 1 and annihilation operator for
m = −1. Treating interaction term as a perturbation, we can say that the “ground state” is the state with l = 0 at each
site, and the (N −fold degenerate) first excited state with l = 1 at some site, which we call a quasi-particle.[1]
Assuming a continuous system, define density correlation function as
C(x, t) =
1
hnα (x, t)nα (0, 0)i
3
(17)
where α denotes the component. In the absence of a magnetic field, (quasi-) particle can be in m = 0, ±1 with equal
probability, and hence
C(x, t) = hn0 (x, t)n0 (0, 0)i
(18)
6
Suppose that (in the Heisenberg picture), we write n0 (x, t) = eiHt n0 (x)e−iHt . Further let K(x, t) denote C(x, t) evaluated
at temperature T = 0. We can interpret this as follows: starting with (say) the ground state, n0 creates a particle at
x = 0, the system then evolves by Newton’s laws for time t. Then, operator n0 (x) acts annihilating a particle at x, after
which we evolve back in time t and take the overlap of this state with the initial state. Thus, we can write,
P
hk|eiHt n0 (x)e−iHt n0 (0)|ki
K(x, t) = k
(19)
Zn
(where Zn is the partition function)
We can depict this pictorially on an x − t plot:
According to our picture, the dotted particle is created and moves while going forward in time. While traversing the
system in reverse time, the new particle has been annihilated. Thus, for non-zero contributino, the trajectories of other
particles in forward time must match for those in reverse time without the extra particle, with the additional criterion
that the particles on the dotted trajectory at times 0 and t must have the same spin. (zero by construction)
5.1
Counting collisions
At time t = 0 the particles’ positions and velocities are as shown, and let 0 denote the position and velocity of the new
particle. Observe that after r right and l left collisions, the particle traveling on the dotted trajectory is the (r − l)th
particle to the right of 0, which is then annihilated at time t. Thus, while evolving in reverse time, the particle which was
at xr−l has been annihilated, and the entire dotted trajectory has been annihilated, and after reverse time evolution, the
corresponding set of trajectories will be in a cyclic permutation. However, the contribution will vanish unless the initial
particle spin and trajectories exactly overlap the final configuration. Thus, we need that all the r − l particles have the
r−l
1
same spin, that has probability
3
Now, each collision contributes a minus sign from the S matrix, however, as nr − nl and nr + nl have the same parity we
can write this contribution as just (−1)nr −nl
7
Hence we can write C(x, t) = hT (C)iK(x, t) where
T (C) =
6
−1
3
nr −nl
(20)
Analytic expression for R(x,t)
For an operator X, the ensemble average over the space of configurations can be calculated as[2]
XZ Y
Y
hXi =
dxν
dvν P ({xν , vν , mν })hxν , vν , mν |X|xν , vν , mν i
mν
ν
(21)
nu
where
"r
#
∆T 2
∆
1 1 Y
− 2c
v
e 2 ν
P ({xν , vν , mν }) = M M
(22)
L 3
2πc2 T
ν
(−1)n
. The average is over the full configuration space defned by the set {xν , vν }M
Our aim is to evaluate R(x, t) =
ν=1 .
3n
C
These represent the position and velocities. Let x(t) denote the ‘dotted’ trajectory. The initial position os fixed, x0 = 0.
Now x(t) = x0 + v0 t. The trajectory ν 0 is given by: xν 0 (t) = xν 0 + vν 0 t.
The probability, P, that these two trajectories have crossed each other at some time, t is goven by:
P = δ [|θ(x − xν 0 (t)) − θ(−xν 0 )| , 1]
So the probability that the ‘dotted’ trajectory has undergone n collissions will be given by:
"
#
X
P (n) = δ
|θ(x − xν 0 (t)) − θ(−xν 0 )| , n
(23)
ν0
We do not know the number of collisions that take place hence we vary n from 0 to ∞ and all the probabilites multiplied
by (−1)n in order to obtain R(x, t). Finally we take the ensemble average of this quantity. Each collision will contribute
to the final integral only if it has the same spin (m = 0) as that of the dotted trajectory. Hence we expect an additional
factor of (1/3)n .
Now a little though convinces us that if we choose the limits of n from −∞ to ∞ and simultaneously remove the
modulous sign from the expression, the expression does not change. Hence we now obtain the final expression for R(x, t).
#+
"
*∞
X (−1)n X
(θ(x − xν 0 (t)) − θ(−xν 0 )) , n
(24)
δ
R(x, t) =
3|n|
−∞
ν0
{xν ,vν }
This form agrees with the second part of equation (86) in Rapp & Zarand.
We introduce the integral repersentation of the Kronecker-delta.
"
# Z
π
X
dφ iφ(n−Pν 0 (θ(x−xν 0 (t))−θ(−xν 0 )))
δ
(θ(x − xν 0 (t) − θ(−xν 0 )) , n =
e
−π 2π
0
(25)
ν
This can be evaluated analytically. Hence we obtain:
R(x, t) =
∞
X
(−1)n
3|n|
−∞
where
Z
π
D(n, x, t) =
−π
D(n, x, t)
dφ inφ
e I(x, t)M
2π
where M is the number of quasiparticles and
D
E
I(x, t) = e−iφ[θ(x−xν 0 (t))−θ(−xν 0 )]
{vν xν }
8
(26)
The integral I(x,t) can be evaluated to be[2]:
t
{G(u) + G(−u) − eiφ G(u) − e−iφ G(−u)}
Lx
I(x, t) = 1 −
where
(27)
2
1
u
G(u) = √ e−u − erfc(u)
2
2 π
where x̄ = x/ζc and t̄ = t/τ . We have defned u = x̄/t̄. ζc and τ are correlation length and time. I(x, t)M can be
re-exponentiated in the large M limit as:
I(x, t)M = e−t̄{G(u)+G(−u)−e
iφ
G(u)−e−iφ G(−u)}
We use the following identity to obtain R(x, t).
∞
X
n=−∞
e
iφn
−1
Q−1
|n|
=
(Q − 1)2 − 1
(Q − 1)2 + 1 + 2(Q − 1) cos φ
(28)
We finally obtain the expression for R(x, t) given by equation (13) in Rapp and Zarand for Q = 4.
(There is a term involving sin(sin(φ)) which vanishes since the function is odd.)
7
Correlation in sine-Gordon model
Suppose we consider action of the sine-Gordon model at low temperature, with solitons acting as quasi-particle excitations.
It can be shown that solitons and anti-solitons follow the same statistic, including the S matrix for collision. It can be
shown[3] that the time dependant correlation functino can be written as
CΦ (x, t) = AhΘs i
(29)
with constant Θ and s being the difference between number of collisions from the right and the left.
Suppose we consider trajectories of quasi-particles defined by their initial position and velocity. Then, in the x − t plot of
trajectories, we can mark “domains” as regions marked out by different trajectories. We can now number the domains by
starting with 0 on the left most, and numbering successive somains as ±1 of the previous domain, depending on whether
the particle trajectory seperating them is a soliton or anti-soliton. Then, the difference in the number of collisions can be
calculated just by the change in the domain number for the domains containing (0, 0) and (x, t) respectively.
The sine-Gordon field, with this representation can be written as
Φ(x, t) =
2π X
mk θ(x − xk (t))
γ
k
where mk is the charge (±1) of the soliton on the trajectory.
9
(30)
Now, to count s = kr − kl , kr and kl being the number of right and left collisions respectively, we observe that the
Z xt
Z ∞
probabilities of right and left collisions are given as qr = ρ
dvP (v)(x − vt)
dvP (v)(x − vt) and ql = ρ
Then, the probability that s = n is simply
X
M →∞
M −kr −kl
− qr − ql )
kr !kl !(M − kr − kl )!
|kr −kl |=n
CΦ (x, t) = lim
x
t
−∞
M !qrkr qlkl (1
X
which gives the correlation function as
M −kr −kl
X
n |kr −kl |=n
M !qrkr qlkl (1 − qr − ql )
kr !kl !(M − kr − kl )!
Θ|kr −kl |
(31)
We now wish to write this partial sum as the coefficient of z n in the Laurent expansion of some function. For n ≥ 0 con
q l M
sider the trinomial expansion of fM (Θz) = 1 − qr − ql + Θzqr +
, and similarly, for n < 0, the trinomial fM (Θ/z)
Θz
M
r
1
qr
√
Θz and b = qr ql , then we can write fM (Θz) = 1 − qr − ql + b w +
. At low temperature, we
Define w =
ql
w
1
can assume that q , q 1 and hence, we can write f (w) ≈ e−(qr +ql ) eb(w+ w )
r
l
M
Now note that in expression (25), the only term explicitly dependant on n is Θ|n| , and hence, summing we get
∞
X
1
Θn =
1−Θ
n=0
Thus, we can write


I
I
−(qr +ql )
1
1
dw
dw
e

q eb(w+ w ) +
q eb(w+ w ) 
CΦ (x, t) = A
q
q
r
r
2πi
Cb w − Θ
Ca w − Θ
ql
(32)
ql
where Ca is a contour enclosing both the singularities anti-clockwise, and Cb is a contour enclosing the origin clockwise.
This can be written in a compactified form in terms of Lommel functions.[3]
8
Dynamical susceptibility and structure factor
The dynamical susceptibility for a system is defined as[4]
Z
χ(q, ω) = −
β
Z
dτ
Z
dx
0
0
dx0 eiωn τ −iq(x−x ) h~n(τ, x)~n(x0 )i|ωn →−iω+δ
(33)
for imaginary time τ
we can write ~n(τ, x) = eHτ ~n(x)e−Hτ
Thus,
Z
χ(q, ω)
β
= −
Z
dτ
Z
dx
0
dx0 eiωn τ −iq(x−x )
0
Z
X
e−βEm e(Em −Ek )τ ~nmk (x)~nkm (x0 )
m,k
β
= −
Z
dτ
Z
dx
dx0
Z
0
0
0
dq 0 e−iq x eiωn τ −iq(x−x )
0
Z
Z
dτ
dq 0 eiωn τ δ(q − q 0 )
0
X
e−βEm e(Em −Ek )τ ñmk (q)ñkm (q 0 )
m,k
β
= −
0
= −
e−βEm e(Em −Ek )τ ~nmk (x)ñkm (q 0 )
m,k
β
= −
Z
X
X
m,k
dτ eiωn τ
X
e−βEm e(Em −Ek )τ ñmk (q)ñkm (−q)
m,k
e−βEm ñmk (q)ñkm (−q)
X
e(Em −Ek )β − 1
e(Em −Ek )β − 1
=−
e−βEm ñmk (q)ñkm (−q)
iωn + Em − Ek
ω − iδ + Em − Ek
m,k
Using the identity
1
lim
=P
δ→0+ x + iδ
10
1
− iπδ(x)
x
(34)
we get
Im χ(q, ω) = − 1 − e−βω
X
e−βEm ñmk (q)ñkm (−q)
(35)
|m−k|=ω
Recall that the dynamical structure factor S(q, ω) is defined as the Fourier transform
Z
S(q, ω)
∞
Z
dx
=
−∞
∞
Z
=
0
Z
dx
Z
dx
0
Z
dx
0
dt eiωt−iq(x−x ) h~n(x, t)~n(x0 )i
0
dt eiωt−iq(x−x )
−∞
Z
dt eiωt
−∞
=
e−βEm ei(Em −Ek )t~nmk (x)~nkm (x0 )
m,k
∞
=
X
X
X
e−βEm ei(Em −Ek )t ñmk (q)ñkm (−q)
m,k
e
−βEm
ñmk (q)ñkm (−q)
(36)
|m−k|=ω
And hence, we can say that dynamical susceptibility and the structure factor are related as
S(q, ω) = − 1 − e−βω Im χ(q, ω)
8.1
(37)
Qualitative form and symmetry of structure factor
Consider the structure factor S(k, ω) for constant k = 0 as a function of ω. Then, assuming Maxwell-Boltzmann distriω
bution the function will have a peak for
= 1 or ω. Note that the explicit espression for R(x, t) given in equation (13)
∆
ref[2] is even in both x, t and hence, it’s Fourier transform will also be real and even in k, ω.
We know that the free particle propogator for this case can be written as[1]
Z
dp
K(x, t) =
D(p)eipx−iε(p)t
(38)
2π
Ac
and ε(p) is the distribution given by (16)
2ε(p)
By the convolution theorem, if C(x, t) = R(x, t)K(x, t) then in Fourier space, S(q, ω) = R̃(q, ω) ∗ K̃(q, ω), that is,
Z
S(q, ω) = dq 0 dω 0 R̃(q − q 0 , ω − ω 0 )D(q 0 )δ(ω 0 − ε(q 0 ))
where D(p) =
(39)
Thus, we can see that despite the Fourier transforms R̃ and K̃ being real and symmetric, their convolution S(q, omega)
ω
= 1. Further, while calculating the dynamical susceptibilty, we have the additional
is not symmetric about the value
∆
−βω
asymmetry factor from 1 − e
. Thus, we can qualitatively see that two sources contribute to the asymmetry of Imχ,
the asymmetry in S(q, ω) itself, and the one added by the 1 − e−βω factor.
9
Conclusion
In this work, we have briefly discussed the semiclassical treatment of quasiparticle excitations in spin one antiferromagnetic
chain with rotor model as the basis. We derived the analytic expression for the spin-spin correlation function, R(x, t),
in analogy with the method done Rapp and Zarand. The central finding is the remarkable correlation in Semiclassical
dynamics of antiferromagnets, paramagnets & ferromagnets [2] and Sine-Gordon Model[3], which results in the same form
of the relaxation function, R(x, t).
This is indeed a very complicated function, which is difficult to evaluate analytically. We generated R(x, t) numerically,
which matched very well with the exact evaluation of R(x, t) at large t. It would be of further interest to numerically
fourier transform and obtain S(k, w). In [4], S(k, w) has been obtained using quantum treatment. Lineshape for S has
been calculated, and an asymmetry has been observed which increases with temperature. This has been claimed to be
a pure quantum mechanical effect unobserved by the semiclassical treatment. It would of interest to know whether the
exact semiclassical treatment predicts any asymmetry or not.
11
References
[1] Damle, Kedar and Sachdev, Subir (1998), Spin dynamics and transport in gapped one-dimensional Heisenberg antiferromagnets at nonzero temperatures, Phys. Rev. B, 57, 14
[2] Rapp, Ákos and Zaránd, Gergely (2006), Dynamical correlations and quantum phase transition in the quantum Potts
model, Phys. Rev. B, 74, 1
[3] Damle, Kedar and Sachdev, Subir (2005), Universal Relaxational Dynamics of Gapped One-Dimensional Models in
the Quantum Sine-Gordon Universality Class, Phys. Rev. Lett., 18, 95
[4] Essler, Fabian H. L. and Konik, Robert M. (2008), Finite-temperature lineshapes in gapped quantum spin chains, Phys.
Rev. B, 78, 10
[5] van Hove, Léon (1954) Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting
Particles Phys. Rev., 95, 1
[6] Sachdev, Subir (1993), Quantum Antiferromagnets in Two Dimensions, arXiv:cond-mat/9303014
12