Download Non-interacting Spin-1/2 Particles in Non

EJTP 6, No. 20 (2009) 243–256
Electronic Journal of Theoretical Physics
Non-interacting Spin-1/2 Particles in
Non-commuting External Magnetic Fields
Kunle Adegoke∗
Physics Department, Obafemi Awolowo University, Ile-Ife, Nigeria
Received 6 July 2008, Accepted 13 January 2009, Published 20 February 2009
Abstract: We obtain, in one dimension, all the energy levels of a system of non-interacting
spin-1/2 particles in non-commuting external magnetic fields. Examples of how to incorporate
interactions as perturbations are given for the Ising model in two orthogonal fields and for the
XZ model in two orthogonal fields.
c Electronic Journal of Theoretical Physics. All rights reserved.
Keywords: Quantum Spin Systems; Ising Systems; Magnetic Fields; Nonlinear Dynamics and
Nonlinear Dynamical Systems; Magnetic Anisotropy; Numerical Simulation Studies
PACS (2008): 75.10.Hk; 75.30.Ds; 05.45.-a
1.
Introduction
Hamiltonian spin models involving two external non-commuting magnetic fields are being
increasingly studied these days [1, 2, 3]. It is almost always the case that such models
cannot be solved exactly, in the presence of spin interactions. Interaction is usually due
to either nearest neighbour exchange or next nearest neighbour exchange or both. As is
common practice, approximate calculations can be done by first starting with an exactly
solvable model and then introducing the interactions as perturbations. This is done in
this paper. We start by considering the simple model described by the Hamiltonian
H = −hx
N
Six − hz
i=1
N
Siz ,
(1)
i=1
where Six and Siz are spin-1/2 operators on the ith lattice site of a one-dimensional chain
of N particles. hx and hz are the external transverse and longitudinal magnetic fields,
respectively, measured in units where the splitting factor and Bohr magneton are unity.
∗
[email protected] and [email protected]
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
244
It will often be convenient to write
H=
N
Hi ,
i=1
where
Hi = −hx Six − hz Siz .
The model (1) is exactly solvable and in this work we will obtain all the energy levels
and the corresponding states. The main import of the result will be that interactions can
be included as perturbations on the model (1), this approach being effective in obtaining
accurate results for models incorporating two external fields.
2.
The One-particle Model
Since the N spin-1/2 particles described by (1) are non-interacting, all results can be
obtained from the Hamiltonian for a single particle. We drop the site subscripts in (1)
and write Hε for the one spin system and write
Hε = −hx S x − hz S z ,
where S x and S z are simply the spin-1/2 operators
⎛
⎞
⎜0
S x = 12 ⎝
1
⎛
1⎟
⎠ , Sz =
0
1
2
⎞
⎜1 0 ⎟
⎝
⎠
0 −1
in a basis with S z diagonal and with = 1. We can denote the basis states in the S z
basis by the set {|↑ , |↓} and in the S x basis by {|→ , |←}, so that
S z |↑ = 12 |↑
S z |↓ = − 12 |↓
S x |→ = 12 |→
S x |← = − 12 |← .
S x is obtained from S z by the similarity transformation
S x = P S z P −1 ,
where P is the unitary matrix
⎛
P =
√1
2
⎞
⎜1 1 ⎟
−1
⎝
⎠=P .
1 −1
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
245
Thus, the S x basis states and the S z basis states are related by
√1
2
|→ =
|← =
√1
2
(|↑ + |↓)
(|↑ − |↓) ,
with the inverse relation
|↑ =
|↓ =
√1
2
√1
2
(|→ + |←)
(|→ − |←) .
The Hamiltonian of the one particle system in the S z basis is
⎛
⎞
⎜ hz hx ⎟
Hε = − 12 ⎝
⎠.
hx −hz
The normalized eigenstates of Hε are
hx |↓ + hz + h2x + h2z |↑
|ε0 = 2 ,
h2x + hz + h2x + h2z
with eigenenergy
ε0 = −
h2x + h2z
,
2
and
hx |↓ + hz − h2x + h2z |↑
|ε1 = 2 ,
h2x + hz − h2x + h2z
with eigenenergy
h2x + h2z
.
ε1 = +
2
Now that we have obtained the eigenstates and the corresponding energies of the
one-particle Hamiltonian, we return to the general Hamiltonian (1).
3.
Energy Levels and Degeneracies
From the results of the previous section and the fact that the particles are non-interacting,
it is clear that there are N + 1 energy levels. We obtain them presently.
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
246
3.1 Ground State
The ground state |E0 of the system of non-interacting N spin-1/2 particles is the direct
product state
|E0 = |ε0 |ε0 |ε0 · · · · · · · · · · · · |ε0 = (|ε0 )N .
(2)
The ground state energy of the system is found as follows
Hi |E0 H |E0 =
i
= (H1 |ε0 ) |ε0 |ε0 · · · · · · · · · · · · · · · |ε0 + |ε0 (H2 |ε0 ) |ε0 · · · · · · · · · · · · |ε0 + |ε0 |ε0 (H3 |ε0 ) |ε0 · · · · · · · · · |ε0 +····································
+ |ε0 |ε0 |ε0 · · · · · · · · · |ε0 (HN |ε0 ) .
Since Hi |ε0 = ε0 |ε0 and there are N terms in the above sum, we thus find that
H |E0 = Nε0 |E0 .
The ground state energy of the model is therefore
E0 = Nε0
N
2
=−
hx + h2z .
2
We shall have more to say about the ground state, in the next section, but presently
we obtain the other energy levels.
3.2 Excited States
A first excited state (FES) of the model is
|E1 1 = |ε0 |ε0 |ε0 · · · · · · · · · |ε0 |ε1 .
We see at once that the FES is N−fold degenerate, since |ε1 can occur anywhere in the
direct product state. The superscript 1 was affixed in anticipation. The remaining N − 1
states that are degenerate with |E1 1 are
|E1 2 = |ε0 |ε0 · · · · · · · · · |ε0 |ε1 |ε0 |E1 3 = |ε0 |ε0 · · · · · · · · · |ε1 |ε0 |ε0 ························
|E1 N = |ε1 |ε0 |ε0 · · · · · · · · · |ε0 |ε0 .
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
247
The energy of the first excited state is found from
H |E1 1 =
i
Hi |E1 1
= (H1 |ε0 ) |ε0 |ε0 · · · · · · · · · |ε0 |ε1 + |ε0 (H2 |ε0 ) |ε0 · · · · · · |ε0 |ε1 +···························
+ |ε0 |ε0 · · · · · · · · · |ε0 (HN |ε1 )
= (N − 1)ε0 |E1 1 + ε1 |E1 1
= [(N − 1)ε0 + ε1 ] |E1 1 .
From which it follows that
E1 = (N − 1)ε0 + ε1
= (N − 2)ε0 .
In general, a kth excited state has k |ε1 factors in the direct product state. Since
there are (N, k) ways of arranging the k |ε1 factors, this means that the kth excited state
has degeneracy g (Ek ), given by
⎛ ⎞
N!
⎜N ⎟
.
g (Ek ) = ⎝ ⎠ =
k!(N
−
k)!
k
The degenerate energy Ek is easily found by the above scheme to be
Ek = (N − k)ε0 + kε1
= (N − 2k)ε0 .
We see that only the ground state |E0 and the Nth excited state |EN are nondegenerate.
4.
An Explicit Expression for the Ground State
We recall from equation (2) that the ground state is the direct product state
|E0 = (|ε0 )N ,
where
hx |↓ + hz + h2x + h2z |↑
|ε0 = 2 .
2
2
2
hx + hz + hx + hz
(3)
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248
Performing the binomial expansion suggested by equation (3), we have that
N
|E0 =
−m
hN
x
m=0
h2x
m
2
2
hz + hx + hz
|Sm 2
+ hz + h2x + h2z
N /2
,
(4)
where |Sm is the linear combination of the (N, m) states with m spins up (in the
total S z basis), that is, the linear combination of all states with total S z = m − N/2.
We can check the trivial limits of (4):
(1) hz = 0
H = −hx
Six
i
Putting hz = 0 in (4) gives
N
|E0 =
m=0
−m m
hN
hx |Sm x
(2h2x )N/2
N
1 = √
|Sm ( 2)N m=0
1
= √
(|↑ + |↓)N
N
( 2)
= (|→)N
= |→→→ · · · · · · →
(2) hx = 0
that is
H = −hz
Siz
i
Here the ground state is trivially the all spins up, non-degenerate ferromagnetic
state
|E0 = |↑↑↑ · · · · · · ↑
with energy
E0 = −Nhz /2
Putting hx = 0 in in (4), we see that the only non-vanishing term in the sum occurs
when m = N and so,
(2hz )N
|SN (2hz )N
= |↑↑↑ · · · · · · ↑ .
|E0 =
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
5.
249
Example Applications
In this section, we will consider two models with interaction, the interactions will be
treated as perturbations. Specifically we will obtain the ground state energy of the Ising
model in two fields and also the XZ model in non-commuting fields to second order in
nearest neighbour exchange interactions. We will also compute first order corrections to
the energy of the first excited state for each model.
We first note that for the one-particle system
2 + h2 |↑
|↓
+
h
+
h
−h
x
z
x
z
1
Sz |ε0 =
2
2
h2x + hz + h2x + h2z
= |α
and
2
2
1 −hx |↓ + hz − hx + hz |↑
Sz |ε1 =
2
2
h2x + hz − h2x + h2z
= |β ,
so that we have the matrix elements
ε0 | S z |ε0 = ε0 | α = 12 √ h2z
hx +h2z
ε1 | S z |ε0 = ε1 | α = − 12 √ h2x
hx +h2z
= ε0 | β
ε1 | S z |ε1 = ε1 | β = − 12 √ h2z
hx +h2z
Furthermore,
and
.
2 + h2 |↓
|↑
+
h
+
h
h
z
x
z
1 x
Sx |ε0 =
2
2
h2x + hz + h2x + h2z
2 + h2 |↓
|↑
+
h
−
h
h
z
x
z
1 x
Sx |ε1 =
2 ,
2
2
2
2
hx + hz − hx + hz
so that we have also the following marix elements:
ε0 | S x |ε0 = ε0 | λ = 12 √ h2x
hx +h2z
ε1 | S x |ε0 = ε1 | λ = 12 √ h2z
hx +h2z
ε1 | S x |ε1 = ε1 | δ = − 12 √ h2x
= − ε1 | α
= ε0 | S x |ε1 = ε0 | δ
hx +h2z
= ε1 | α = − ε0 | λ
(5)
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250
We are now ready to compute the energy corrections.
5.1 Perturbation Expansion of the Ground State Energy of the Ising
Model in Non-Commuting Fields
The Ising model in non-commuting magnetic fields is described by the Hamiltonian
HI = −j
z
Siz Si+1
− hx
i
Six − hz
i
Siz
i
=V +H,
where
V = −j
(6)
z
Siz Si+1
i
and H is as given in (1).
The nearest neighbour exchange interaction j is assumed positive, so that we have a
ferromagnetic model. Furthermore, we assume a periodic boundary condition, so that
z
z
SN
+i = Si . For weak interaction, V can be treated as a perturbation.
5.1.1 First Order Energy Correction
The first order correction to the ground state energy of the Hamiltonian (6) is given by
(1)
ΔE0I = E0 | V |E0 z
= −j E0 |
Siz Si+1
|E0 .
i
Now,
i
z
z z
Siz Si+1
|E0 = S1z S2z |E0 + S2z S3z |E0 + · · · + SN
S1 |E0 = |α |α |ε0 |ε0 · · · · · · · · · |ε0 + |ε0 |α |α |ε0 · · · · · · |ε0 + |ε0 |ε0 |α |α · · · · · · |ε0 +························
+ |α |ε0 |ε0 · · · · · · |ε0 |α .
Multiplying from the left by E0 |, we have
z
Siz Si+1
|E0 = N (
ε0 | α)2
E0 |
i
=
N h2z
.
4 h2x + h2z
(7)
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251
We therefore have that the first order correction to the ground state energy of the Ising
model in non-commuting magnetic fields is
(1)
ΔE0I =
−jN h2z
.
4 h2x + h2z
5.1.2 Second Order Correction
The second order correction to the ground state energy of the Ising model in noncommuting fields has the form
(2)
ΔE0I =
|
E| V |E0 |2
.
E0 − E
E
(8)
From the form of (7) and the fact that ε0 | ε1 = 0 it is clear that only states with
E = E1 and those with E = E2 can contribute to the sum in (8). We consider them in
turns.
When E = E1 , we observe that |E1 1 = |ε0 |ε0 · · · · · · |ε0 |ε1 gives
1
E1 |
z
Siz Si+1
|E0 i
= ε1 | ε0 | ε0 | · · · · · · ε0 | ε0 |ε0 |ε0 · · · · · · |ε0 |α |α
+ ε1 | ε0 | ε0 | · · · · · · ε0 | α |ε0 |ε0 · · · · · · |ε0 |ε0 |α
= 2 ε1 | α ε0 | α .
We have similar results from the remaining N − 1 states that are degenerate with
|E1 1 . The E = E1 states therefore contribute
(2 ε1 | α ε0 | α)2 (−j)2
Nj 2 h2x h2z
N
=
2ε0
8ε0 (h2x + h2z )2
to the sum in (8). We have used (5) to substitute for ε1 | α and ε0 | α and we
have also used the fact that E0 − Ek = 2kε0.
As for the states with E = E2 , only N (those with two consecutive |ε1 factors) of
the (N, 2) states contribute to the sum. Their contribution is
(
ε1 | α)2 (−j)2
Nj 2
h4x
N
=
.
4ε0
64ε0 (h2x + h2z )2
The second order correction to the ground state energy of the Ising model in noncommuting fields is therefore
(2)
ΔE0I =
h4x
Nj 2 h2x h2z
Nj 2
+
.
8ε0 (h2x + h2z )2 64ε0 (h2x + h2z )2
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252
5.2 Perturbation expansion of the ground state energy of the XZ model
in non-commuting fields.
The XZ model in one dimension, in the presence of non-commuting external fields is
described by the Hamiltonian
Hxz = −j1
N
i=1
x
Six Si+1
− j2
N
i=1
z
Siz Si+1
− hx
N
Six − hz
i=1
N
Siz ,
i=1
where j1 > 0 and j2 > 0 are the nearest neighbour exchange interaction energy. Again
x
x
z
z
we assume a periodic boundary condition, so that SN
+i = Si and SN +i = Si . For weak
nearest neighbour interactions, after calculations similar to that in the previous section,
we obtain the following corrections to the ground state energy:
2
1
N
(1)
2
j
h
+
j
h
ΔE0XZ = −
1
2
x
z
4 h2x + h2z
and
2
N
h2x h2z
(2)
j1 + j22
ΔE0XZ =
2
8ε0 (h2x + h2z )
2 4
1
N
j1 hx + j22 h4z .
+
2
64ε0 (h2x + h2z )
The ground state energy of the XZ model in non-commuting fields, to second order in
interactions j1 and j2 is thus
2
1
N
2
N
hx + h2z −
j1 hx + j2 h2z
E0XZ = −
2
2
2
4 hx + hz
2 2
N
hx hz
2
2
j
+
+
j
2
8ε0 (h2x + h2z )2 1
2 4
1
N
2 4
j
h
+
j
h
+
2 z .
64ε0 (h2x + h2z )2 1 x
5.3 First Order Correction to the Energy of the First Excited State (FES)
of the Ising Model in Non-Commuting Fields
The first excited state, in the absence of interactions, is N−fold degenerate, with the N
states as given in (3.2). We therefore use degenerate perturbation theory to find the first
order correction to the first excited state energy of the Ising model in non-commuting
fields.
z z
Si Si+1 |E1 1 = |α |α |ε0 |ε0 · · · · · · · · · |ε1 i
+ |ε0 |α |α |ε0 · · · · · · |ε0 |ε1 +···························
+ |ε0 |ε0 |ε0 · · · · · · |ε0 |α |β
+ |α |ε0 |ε0 · · · · · · |ε0 |ε0 |β .
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
We therefore have that
1
E1 |
253
z
Siz Si+1
|E1 1
i
= (N − 2) (
ε0 | α)2 + 2 ε0 | α ε1 | β
N
h2z
−1 .
= 2
hx + h2z 4
A similar calculation for the remaining N − 1 states shows that the diagonal elements
of the N × N perturbation matrix V , are equal, being given by:
i
Vii = E1 | −j
=
−jh2z
h2x + h2z
We can also compute V21 thus
V21 =2
E1 | −j
z
Siz Si+1
|E1 i
i
N
−1 =f.
4
(9)
z
Siz Si+1
|E1 1
i
= −j ε0 | ε1 | ε0 | · · · · · · ε0 | ε0 · · · · · · |α |β
= −j ε1 | α ε0 | β
−j h2x
.
=
4 h2x + h2z
In fact, for m = n, we have
Vmn =m
E1 | −j
h2x
z
Siz Si+1
|E1 n
i
−j
δm,n±1
4 h2x + h2z
= Vnm = g .
=
The N × N perturbation matrix V is therefore
⎛
⎜f g 0 0 0 0
⎜
⎜g f g 0 0 0
⎜
⎜
⎜
⎜0 g f g 0 0
⎜
⎜
⎜0 0 g f g 0
⎜
V =⎜
⎜
⎜0 0 0 g f g
⎜
⎜
..
⎜0 0 0
.
⎜
⎜.
⎜ ..
g
⎜
⎝
g 0 0 0 ··· 0
(10)
⎞
··· g ⎟
⎟
··· 0⎟
⎟
⎟
⎟
··· 0⎟
⎟
⎟
··· 0⎟
⎟
.. ⎟
⎟
.⎟
⎟
⎟
⎟
⎟
⎟
f g⎟
⎟
⎠
g f
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254
where f and g are as given in (9) and (10). Clearly f and g are negative for N > 4. The
smallest eigenvalue of V is 2g + f and is non-degenerate, so that the correction to the
first excited state energy, to first order in interaction j is
(1)
ΔE1I = 2g + f
N
−j h2x
jh2z
−1 .
=
−
2 h2x + h2z h2x + h2z 4
The nearest neighbour interactions thus lift the degeneracy in the first excited state
of the model. The energy of the first excited state of the Ising model in non-commuting
fields, to first order in j is therefore
E1I
2
2
N 2
j
jh
N
h
x
z
= 1−
hx + h2z −
−
−1 .
2
2 h2x + h2z h2x + h2z 4
5.4 First Order Correction to the Energy of the First Excited State (FES)
of the XZ Model in Non-Commuting Fields
A completely analogous treatment to the above gives the energy of the first excited state
of the XZ model in non-commuting fields, to first order in the interactions j1 and j2 as
N
−1
=−
h2x + h2z
2
j1 h2x + j2 h2z
N
1 j2 h2x + j1 h2z
−
1
−
−
.
2 h2x + h2z
4
h2x + h2z
E1XY
Conclusion
We have obtained all the energy levels of a system of N non-interacting spin-1/2 particles
in non-commuting external magnetic
in one dimension. The energy of the kth level
fields,
2
is Ek = (N − 2k)ε0, where ε0 = − (hx + h2z )/2 and k = 0, 1, 2, . . . , N. The kth energy
level was found to be (N, k) = N!/[k!(N − k)!]-fold degenerate. An expicit expression for
the ground state
was also derived (equation (4)). It is not difficult to give similar explicit expressions
for the excited states. A scheme for obtaining the energies of the excited states was
highlighted in section 3.2.
Since most real systems do interact, examples of how the interaction terms of an Hamiltonian can be included by Rayleigh-Schrödinger perturbation was demonstrated by finding
energy corrections to the ground state energy of the Ising model in non-commuting fields,
as well as to the ground state energy of the XZ model in non-commuting fields. Corrections to the first excited state energy were also calculated.
Electronic Journal of Theoretical Physics 6, No. 20 (2009) 243–256
255
Acknowledgments
The author is grateful to the DAAD for a scholarship and thanks the Physics Institute,
Universität Bayreuth for hospitality.
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