Download Presentation slides

“Billiard ball” model for structure factor in 1-D
Heisenberg anti-ferromagnets
Shreyas Patankar1
Chennai Mathematical Institute
August 5, 2010
1
Project with Prof. Kedar Damle, TIFR and Girish Sharma, Satyasai U
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
1 / 21
Motivation
Inelastic neutron scattering
Inelastic neutron scattering used to study magnetic solids
Magnetic moment of neutrons acts on spin states
Apparatus to measure
q = p1 − p2 (change in momentum)
(1)
∆ = E1 − E2 (change in energy)
(2)
Experiment used to calculate the “structure factor” S(q, ω)
(ω gives measure of energy)
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
2 / 21
Motivation
Plan
We have 1-D Heisenberg spin S = 1 chain with antiferromagnetic
interaction
(Nearest neighbour interaction)
Use fancy math to simplify Hamiltonian to quantum rotor model
“Count” to get two point correlation function analytically
Fourier transform to give dynamical structure factor S(k, ω)
Throughout we use units with ~ = kB = 1
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
3 / 21
Motivation
Plan
We have 1-D Heisenberg spin S = 1 chain with antiferromagnetic
interaction
(Nearest neighbour interaction)
Use fancy math to simplify Hamiltonian to quantum rotor model
“Count” to get two point correlation function analytically
Fourier transform to give dynamical structure factor S(k, ω)
Throughout we use units with ~ = kB = 1
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
3 / 21
Motivation
Plan
We have 1-D Heisenberg spin S = 1 chain with antiferromagnetic
interaction
(Nearest neighbour interaction)
Use fancy math to simplify Hamiltonian to quantum rotor model
“Count” to get two point correlation function analytically
Fourier transform to give dynamical structure factor S(k, ω)
Throughout we use units with ~ = kB = 1
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
3 / 21
Motivation
Plan
We have 1-D Heisenberg spin S = 1 chain with antiferromagnetic
interaction
(Nearest neighbour interaction)
Use fancy math to simplify Hamiltonian to quantum rotor model
“Count” to get two point correlation function analytically
Fourier transform to give dynamical structure factor S(k, ω)
Throughout we use units with ~ = kB = 1
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
3 / 21
Motivation
Plan
We have 1-D Heisenberg spin S = 1 chain with antiferromagnetic
interaction
(Nearest neighbour interaction)
Use fancy math to simplify Hamiltonian to quantum rotor model
“Count” to get two point correlation function analytically
Fourier transform to give dynamical structure factor S(k, ω)
Throughout we use units with ~ = kB = 1
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
3 / 21
The model Hamltonian
Spin system - path integral
Consider 1-D system of spins (S = 1) with Hamiltonian
X
~i · S
~j
H=J
S
(3)
hiji
with J > 0
We can write partition function for system as a path integral
Z
Rβ
−βH
Z = Tr e
=
Dα(τ ) e− 0 S[α(τ )]dτ
(4)
α(0)=α(β)
where |α(τ )i are states in the Hilbert space, τ is parameter called
“imaginary time” and
S[α(τ )] = lim
→0
Shreyas Patankar (CMI)
hα(τ + )|βH|α(τ )i
Spin-1 Heisenberg antiferromagnets
August 5, 2010
4 / 21
The model Hamltonian
Coherent state representation
Let us write |αi states in more commonly used notation:
~ i = e−iφŜ3 e−iθŜ2 e−iψŜ3 | ↑i
|N
(5)
notice the Euler rotation angles
Satisfies the property that
~ |S|
~N
~ i = SN
~
hN
(6)
~ is the unit vector corresponding to the coherent state
where N
Thus partition function can be written as
hR
i
Z
R
~ (τ )| d |N
~ (τ )idτ + β H(S N
~ (τ ))
− 0β hN
0
dτ
~
Z=
DN (τ ) e
(7)
~ (0)=N
~ (β)
N
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
5 / 21
The model Hamltonian
Berry Phase
Observe that the first term in exponent is purely imaginary
Z β
∗
Z β
d ~
d ~
~
~
dτ hN (τ )| |N (τ )i = −
dτ hN (τ )| |N (τ )i
dτ
dτ
0
0
This term is known as the Berry phase
~ (τ )|β
It gives surface area of part of sphere bound by the path N
0
This can be re-written as
Z β Z
SB = iS
dτ
0
1
~ (u, τ ) ·
duN
0
~
~
dN
dN
×
du
dτ
!
(8)
with u as additional parameter
Observe: Berry phase has nothing to do with Hamiltonian
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
6 / 21
The model Hamltonian
Long distance effective model
Approximation: antiferromagnetic “orientation” vector varies
gradually.
Split Hamiltonian as
r
a2d ~ 2
ad ~
j
~
N (xj , τ ) = (−1) ~n(xj , τ ) 1 − 2 L
+
L(xj , τ )
S
S
(9)
~ and |L|
~ is small
where ~n is slowly varying, ~n ⊥ L
~ as the total angular momentum density
We can think of L
Substituting, we can write Z as
Z
Rβ
R d ρs
~2
2
2 L
~ (τ ) eSB − 0 dτ d x 2 (∂τ n̂) +S 2χ⊥
Z=
DN
(10)
~ (0)=N
~ (β)
N
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
7 / 21
The model Hamltonian
Quantum Rotor Model
Quantum rotor model
Re-discretizing we can write
X gL2 J
i
H=
+ (~ni+1 − ~ni )2
2
2
(11)
i
Hamiltonian known as the Rotor model
Think of it as chain of rigid rotors, interaction favoring
nearest-neighbour alignment
Can be shown that even for small values of g, first term dominates
the Hamiltonian
Y
Eigenstates of the unperturbed Hamiltonian are
|li , mi i
i
where li , mi are eigenvalues of L2i , Li;z respectively
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
8 / 21
The model Hamltonian
Quantum Rotor Model
Perturbation
Rewrite Hamiltonian as
H=
X gL2
J
+ ~ni · ~ni+1
2
2
i
i
Treat ~ni · ni+1
~ term as a perturbation
Ground state is all sites with l = 0
(Degenerate) first excited state has l = 1, m = 0, ±1 for one site
Need to use degenerate perturbation
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
9 / 21
The model Hamltonian
Quantum Rotor Model
Dispersion relation
1 X ikaq
To diagonalize, choose basis |ki = √
e
|qi
N q
q is label for first excited states
Expectation value gives energy as function of k
We have dispersion relation
ε(k) = E1 −
2J
cos(ka)
3
(12)
we can call k as “momentum”
c2 k 2
∆
For small k, ε(k) = ∆ +
+ . . ., identify 2 as mass
2∆
c
We have excitations as “quasiparticles”
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
10 / 21
The model Hamltonian
Quantum Rotor Model
Quasi-particles
Each quasi-particle has l = 1 (by definition) and comes in three
“colors”: m = 0, ±1
Let ~n be the rotor position (vector) operator
Define
n̂0 = n̂z
n̂+ = n̂x + in̂y
n̂− = n̂x − in̂y
We can show[1] that n̂0 is sum of creation and annihilation operators
for m = 0 quasiparticles
Thus, n̂0;i |0i will produce a particle of m = 0 at site i
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
11 / 21
Structure factor
Two-point correlation function
We wish to calculate structure factor
It can be shown[5] that structure factor is Fourier transform of two
point correlation function
ZZ ∞
S(k, ω) =
dxdt C(x, t)e−i(kx+ωt)
−∞
where C(x, t) = hn̂(x, t)n̂(0, 0)i
n̂(x, t) can be thought of as operator acting at (x, t)
Write (Heisenberg picture)
n0 (x, t) = e−iHt n0 (x, 0)eiHt
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
12 / 21
Semi-classical approximation
Quasi-particle collision
Assume small temperature: T ∆ we have small number of
excitations
It can be shown[1] that S matrix for quasi-particle collision is just −1
We can visualize C(x, t) = he−iHt n0 (x, 0)eiHt n̂(0, 0)i as
1
2
3
4
Creation of quasi particle at (0,0)
Time evolution of system by t
Annihilation of particle at (x,t)
Reverse time evolution of background by t
For non-zero contribution, initial background state must match with
final background state
Further, created and annihilated particle must have same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
13 / 21
Semi-classical approximation
Quasi-particle collision
Assume small temperature: T ∆ we have small number of
excitations
It can be shown[1] that S matrix for quasi-particle collision is just −1
We can visualize C(x, t) = he−iHt n0 (x, 0)eiHt n̂(0, 0)i as
1
2
3
4
Creation of quasi particle at (0,0)
Time evolution of system by t
Annihilation of particle at (x,t)
Reverse time evolution of background by t
For non-zero contribution, initial background state must match with
final background state
Further, created and annihilated particle must have same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
13 / 21
Semi-classical approximation
Quasi-particle collision
Assume small temperature: T ∆ we have small number of
excitations
It can be shown[1] that S matrix for quasi-particle collision is just −1
We can visualize C(x, t) = he−iHt n0 (x, 0)eiHt n̂(0, 0)i as
1
2
3
4
Creation of quasi particle at (0,0)
Time evolution of system by t
Annihilation of particle at (x,t)
Reverse time evolution of background by t
For non-zero contribution, initial background state must match with
final background state
Further, created and annihilated particle must have same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
13 / 21
Semi-classical approximation
Quasi-particle collision
Assume small temperature: T ∆ we have small number of
excitations
It can be shown[1] that S matrix for quasi-particle collision is just −1
We can visualize C(x, t) = he−iHt n0 (x, 0)eiHt n̂(0, 0)i as
1
2
3
4
Creation of quasi particle at (0,0)
Time evolution of system by t
Annihilation of particle at (x,t)
Reverse time evolution of background by t
For non-zero contribution, initial background state must match with
final background state
Further, created and annihilated particle must have same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
13 / 21
Semi-classical approximation
Quasi-particle collision
Assume small temperature: T ∆ we have small number of
excitations
It can be shown[1] that S matrix for quasi-particle collision is just −1
We can visualize C(x, t) = he−iHt n0 (x, 0)eiHt n̂(0, 0)i as
1
2
3
4
Creation of quasi particle at (0,0)
Time evolution of system by t
Annihilation of particle at (x,t)
Reverse time evolution of background by t
For non-zero contribution, initial background state must match with
final background state
Further, created and annihilated particle must have same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
13 / 21
Semi-classical approximation
Quasi-particle collision
Assume small temperature: T ∆ we have small number of
excitations
It can be shown[1] that S matrix for quasi-particle collision is just −1
We can visualize C(x, t) = he−iHt n0 (x, 0)eiHt n̂(0, 0)i as
1
2
3
4
Creation of quasi particle at (0,0)
Time evolution of system by t
Annihilation of particle at (x,t)
Reverse time evolution of background by t
For non-zero contribution, initial background state must match with
final background state
Further, created and annihilated particle must have same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
13 / 21
Semi-classical approximation
Collision contd.
We can depict pictorially as x − t plot
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
14 / 21
Semi-classical approximation
Trajectories
Semiclassically, think of quasi-particles having definite trajectories
In ensemble average for C(x, t), each state is a “configuration” of
trajectories
Dotted trajectory is one created by action of n̂0
Suppose dotted trajectory has r right and l left collisions
Then annihilated particle is |r − l| positions to right of created particle
For solid lines to be reversal invariant, all intermediate particles must have
same m = 0
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
15 / 21
Semi-classical approximation
Response function
Suppose we take temperature T = 0
Then C(x, t) is just the free (quasi-)particle propogator K(x, t)[2]
At finite temperature, there is term due to collisions (S-matrix and
probability)
|r−l|
1
r+l
From above observations, these contributions are (−1) ,
3
respectively
Thus, we have
*
where R(x, t) =
C(x, t) = R(x, t)K(x, t)
+
−1 |r−l|
3
(13)
C
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
16 / 21
Semi-classical approximation
Counting collisions
Suppose configuration of trajectories is given by xν = xν + vν t and
dotted trajectory is x = x0 + v0 t
Probability that there are n collisions on this trajectory is:
"
#
X
P (n) = δ
|θ(x − xν (t)) − θ(−xν )| , n
(14)
ν
This gives
#+
"
∞
X
(−1)n X
δ
(θ(x − xν 0 (t)) − θ(−xν 0 )) , n
3|n|
0
n=−∞
*
R(x, t) =
ν
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
{xν ,vν }
August 5, 2010
17 / 21
Semi-classical approximation
Expression for R(x, t)
Expression can be simplified(!) to give[2]:
Z
π
R(x̄, t̄) =
−π
dφ −|t|(1−cos φ)
e
2π
2
√1 e−u +u
π
erf(u)
where x̄, t̄ are in suitably normalized units, u =
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
cos(x̄ sin φ)
4
5 + 3 cos φ
x̄
t̄
August 5, 2010
18 / 21
Conclusion
Features
Pretty picture:
Figure: Real part of C(x, t) vs. x, t
Observe interesting behaviour over t
Analytic expression itself gives assymetry in S(q, ω) about q
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
19 / 21
Conclusion
Comparison
Essler, Konik calculated S(q, ω) using NLσM field theory methods
Figure: Susceptibility vs. frequency at different temperatures
It is claimed that the assymetry is a purely quantum effect and
cannot be reproduced semi-classically
Task: To verify this claim!
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
20 / 21
References
References
Damle, Kedar; Sachdev, Subir (1998),
Phys. Rev. B, 57, 14
Rapp, Ákos; Zaránd, Gergely (2006),
Phys. Rev. B, 74, 1
Damle, Kedar; Sachdev, Subir (2005),
Phys. Rev. Lett., 18, 95
Essler, Fabian H. L.; Konik, Robert M. (2008),
Phys. Rev. B, 78, 10
van Hove, Léon (1954)
Phys. Rev., 95, 1
Sen, Arnab (2007),
Deconfined Quantum Critical Points (MSc. thesis)
Shreyas Patankar (CMI)
Spin-1 Heisenberg antiferromagnets
August 5, 2010
21 / 21