Spin Hamilton Operators, Symmetry Breaking, Energy Level Crossing and Entanglement

0.08 MB·English

by  Willi-Hans Steeb

Spin Hamilton Operators, Symmetry Breaking, Energy Level Crossing
and Entanglement
Willi-Hans Steeb, Yorick Hardy and Jacqueline de Greef
International School for Scientific Computing,
2
University of Johannesburg, Auckland Park 2006, South Africa,
1
0 e-mail: [email protected]
2

n
a
J

7
2
Abstract We study finite-dimensional product Hilbert spaces, coupled spin sys-

] tems, entanglement and energy level crossing. The Hamilton operators are based
h
p on the Pauli group. We show that swapping the interacting term can lead from un-
-
t entangled eigenstates to entangled eigenstates and from an energy spectrum with
n
a energy level crossing to avoided energy level crossing.
u
q
[

2
v 1 Introduction
4
0
2 Let 1, 2 be Hilbert spaces and 1 2 be the tensor product Hilbert space
4 H H H ⊗ H
. [1,2]. Quiteoftena self-adjoint Hamiltonoperatoracting onthethe tensor product
0
1 Hilbert space 1 2 can be written as
1 H ⊗H
1
v: Hˆ = Hˆ1 I2 +I1 Hˆ2 +ǫVˆ (1)
⊗ ⊗
i
X
r where the self-adjoint Hamilton operator Hˆ1 acts in the Hilbert space 1, the
a H
self-adjoint Hamilton operator Hˆ2 acts in the Hilbert space 2, I1 is the iden-
H
tity operator acting in the Hilbert space 1 and I2 is the identity operator acting
H
in the Hilbert space 2. The self-adjoint operator Vˆ acts in the product Hilbert
H
space andǫ isa real parameter. The main taskwould beto findthe spectrum ofHˆ.
In the following we consider the finite-dimensional Hilbert space 1 = 2 = Cn
H H
and then denotes the Kronecker product [3,4,5,6]. Let I be the n n identity
n
⊗ ×
matrix. We consider the two hermitian Hamilton operators
Hˆ = αA I +I βB +ǫ(A B) (2)
n n
⊗ ⊗ ⊗
Kˆ = αA I +I βB +ǫ(B A) (3)
n n
⊗ ⊗ ⊗
where A, B are nonzero n n hermitian matrices and α, β, ǫ are real parameters
×
with ǫ 0. We assume that [A,B] = 0. The vector space of the n n matrices
≥ 6 ×
over C form a Hilbert space with the scalar product X,Y := tr(XY∗). We also
h i
assume that A,B = 0, i.e. the nonzero n n hermitian matrices A and B are
h i ×
orthogonal to each other. Of particular interest would be the case where A and
B are elements of a semi-simple Lie algebra. We discuss the eigenvalue problem
for the two Hamilton operators and its connection with entanglement and energy
level crossings for specific choices of A and B. In the following the matrices A
and B are realized by Pauli spin matrices. The Hamilton operator will be a linear
combination of elements of the Pauli group . The Pauli group [7] is defined by
n
P
n := I2, σx, σy, σz ⊗n 1, i . (4)
P { } ⊗{± ± }
Suchtwo-level andhigherlevelquantumsystemsandtheirphysical realizationhave
beenstudiedbymanyauthors(see[8]andreferences therein). Thethermodynamic
behaviour is determined by the partition functions
Zˆ(β) = tr(exp( βHˆ)), Zˆ(β) = tr(exp( βKˆ)).
H − K −
Since
tr(Hˆ) = tr(Kˆ) = αntr(A)+βntr(B)+ǫ(tr(A))(tr(B))
the sum of the eigenvalues of the operators Hˆ and Kˆ are the same. However, in
general, the partition functions will be different.
2 Commutators, Eigenvalues and Eigenvectors
Let us first summarize the equations we utilize in the following. Let A, B be n n
×
matrices over C. First note that we have the following commutators
[A I ,I B] = 0, [A I ,A B] = 0, [I B,A B] = 0
n n n n
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
and
[A I ,B A] = ([A,B]) A, [I B,B A] = B ([B,A]).
n n
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
Thelasttwocommutatorswouldbe0if[A,B] = 0. Thereisann2 n2 permutation
×
matrix P (swap gate) such that
P(A B)P−1 = B A.
⊗ ⊗
This implies that P(A I )P−1 = I A and P(I B)P−1 = B I .
n n n n
⊗ ⊗ ⊗ ⊗
Now let A and B be n n hermitian matrices. If the eigenvalues and normalized
×
eigenvectors of A and B are λ ,u , µ ,v , (j = 1,2,...,n), respectively, then the
j j j j
eigenvalues and normalized eigenvectors of the Hamilton operator (2) are given by
[3,4,5,6]
αλ +βµ +ǫλ µ , u v j,k = 1,2,...,n.
j k j k j k
⊗
Thustheeigenvectors arenotentangledsincetheycanbewrittenasproductstates.
These results can be extended to the Hamilton operator
Hˆ = α(A I I )+β(I B I )+γ(I I C)+ǫ(A B C)
n n n n n n
⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗ ⊗
and higher dimensions.
3 Pauli Spin Matrices and Entanglement
Since we realize the linear operators A and B by Pauli spin matrices we summarize
some results for the Pauli spin matrices and their Kronecker products. Consider
the Pauli spin matrices σ , σ , σ . The eigenvalues are given by +1 and 1 with
z x y
−
the corresponding normalized eigenvectors
1 0 1 1 1 1 1 i 1 i
, , , , − ,
(cid:18)0(cid:19) (cid:18)1(cid:19) √2 (cid:18)1(cid:19) √2 (cid:18) 1(cid:19) √2 (cid:18) 1 (cid:19) √2 (cid:18)1(cid:19)
−
for σ , σ and σ , respectively. Consider now the three hermitian and unitary
z x y
4 4 matrices σ σ , σ σ , σ σ . These matrices appear in Mermin’s magic
x x y y z z
× ⊗ ⊗ ⊗
square [9] for the proof of the Bell-Kochen-Specker theorem. Since the eigenvalues
of the Pauli matrices are given by +1 and 1, the eigenvalues of the 4 4 matrices
− ×
σ σ , σ σ , σ σ are +1 (twice) and 1 (twice). The eigenvectors can
x x y y z z
⊗ ⊗ ⊗ −
be given as product states (unentangled states), but also as entangled states (i.e.
they cannot be written as product states). Obviously
1 1 1 0 0 1 0 0
, , ,
(cid:18)0(cid:19)⊗(cid:18)0(cid:19) (cid:18)0(cid:19)⊗(cid:18)1(cid:19) (cid:18)1(cid:19)⊗(cid:18)0(cid:19) (cid:18)1(cid:19)⊗(cid:18)1(cid:19)
are four normalized product eigenstates of σ σ . The normalized product eigen-
z z
⊗
states of σ σ are
x x
⊗
1 1 1 1 1 1 1 1 1 1 1 1
, , , .
2 (cid:18)1(cid:19)⊗(cid:18)1(cid:19) 2 (cid:18)1(cid:19)⊗(cid:18) 1(cid:19) 2 (cid:18) 1(cid:19)⊗(cid:18)1(cid:19) 2 (cid:18) 1(cid:19)⊗(cid:18) 1(cid:19)
− − − −
The normalized product eigenstates of σ σ are
y y
⊗
1 i i 1 i i 1 i i 1 i i
, − , − , − − .
2 (cid:18)1(cid:19)⊗(cid:18)1(cid:19) 2 (cid:18)1(cid:19)⊗(cid:18) 1 (cid:19) 2 (cid:18) 1 (cid:19)⊗(cid:18)1(cid:19) 2 (cid:18) 1 (cid:19)⊗(cid:18) 1 (cid:19)
All three 4 4 matrices also admit the Bell basis
×
1 0 1 0
1 0 1 1 1  0  1  1 
, , ,
√2 0 √2 1 √2 0 √2 1
      − 
1 0  1  0 
    −   
as normalized eigenvectors which are maximally entangled. As measure for entan-
glement the tangle [5,7,10,11] will be utilized.
Consider now the hermitian and unitary 4 4 matrices σ σ , σ σ . Since the
x z z x
× ⊗ ⊗
eigenvaluesofthePaulimatricesaregivenby+1and 1,theeigenvaluesofthe4 4
− ×
matrices σ σ and σ σ , are +1 (twice) and 1 (twice). The eigenvectors can
x z z x
⊗ ⊗ −
be given as product states (unentangled states), but also as entangled states (i.e.
they cannot be written as product states). The normalized product eigenstates of
σ σ are
x z
⊗
1 1 1 1 1 0 1 1 1 1 1 0
, , , .
√2 (cid:18)1(cid:19)⊗(cid:18)0(cid:19) √2 (cid:18)1(cid:19)⊗(cid:18)1(cid:19) √2 (cid:18) 1(cid:19)⊗(cid:18)0(cid:19) √2 (cid:18) 1(cid:19)⊗(cid:18)1(cid:19)
− −
The normalized product eigenstates of σ σ are
z x
⊗
1 1 1 1 1 1 1 0 1 1 0 1
, , , .
√2 (cid:18)0(cid:19)⊗(cid:18)1(cid:19) √2 (cid:18)0(cid:19)⊗(cid:18) 1(cid:19) √2 (cid:18)1(cid:19)⊗(cid:18)1(cid:19) √2 (cid:18)1(cid:19)⊗(cid:18) 1(cid:19)
− −
The two 4 4 matrices also admit
×
1 1 1 1
− −
1  1 1  1  1  1 1  1 
− , , − ,
2 1 2 1 2 1 2 1
−      − 
 1   1   1   1 
       
as normalized eigenvectors which are maximally entangled. Note that σ σ also
y y
⊗
admits these maximally entangled eigenvectors besides the Bell basis as eigenvec-
tors and the product eigenvectors.
For the triple spin interaction term σ σ σ we obtain the eigenvalues +1
x y z
⊗ ⊗
(fourfold) and 1 (fourfold) and all the eight product states as eigenstates given
−
by the eigenstates of σ , σ , σ . Owing to the degeneracies of the eigenvalues we
x y z
also find fully entangled states such as
1
(1 1 0 0 0 0 i i)T
2 −
with the three-tangle as measure [11].
4 Examples
Consider now a specific example for αA and βB with n = 2 and ǫ > 0. Utilizing
the Pauli spin matrices
αA = ~ω1σz, βB = ~ω2σx
where α = ~ω1, β = ~ω2 and ω1, ω2 are the frequencies. Note that [σz,σx] = 2iσy
and tr(Hˆ) = 0, tr(Kˆ) = 0. The elements of the set
I I ,σ I ,I σ ,σ σ
2 2 z 2 2 x z x
{ ⊗ ⊗ ⊗ ⊗ }
form a commutative subgroup of the Pauli group 2. The elements σz I2, I2 σx,
P ⊗ ⊗
σx σz are generators of the Pauli group 2. Now the eigenvalues and eigenvectors
⊗ P
of αA are given by
1 0
λ1 = ~ω1, u1 = , λ2 = ~ω1, u2 =
(cid:18)0(cid:19) − (cid:18)1(cid:19)
and the eigenvalues and eigenvectors of βB are given by
1 1 1 1
µ1 = ~ω2, u1 = , µ2 = ~ω2, u2 = .
√2 (cid:18)1(cid:19) − √2 (cid:18) 1(cid:19)
−
The Hamilton operator Hˆ is given by the 4 4 matrix which can be written as
×
direct sum of two 2 2 matrices
×
~ω1 ~ω2 +ǫ 0 0
~ω2 +ǫ ~ω1 0 0 
H = .
0 0 ~ω1 ~ω2 ǫ
 − − 
e  0 0 ~ω2 ǫ ~ω1 
 − − 
The eigenvalues of H are
E1(ω1,ω2,ǫ) e= ~ω1 +~ω2 +ǫ, E2(ω1,ω2,ǫ) = ~ω1 ~ω2 ǫ
− −
E3(ω1,ω2,ǫ) = ~ω1 +~ω2 ǫ, E4(ω1,ω2,ǫ) = ~ω1 ~ω2 +ǫ
− − − −
with the corresponding eigenvectors (which can be written as product states)
1 1 1 1 1 1
, ,
(cid:18)0(cid:19)⊗ √2 (cid:18)1(cid:19) (cid:18)0(cid:19)⊗ √2 (cid:18) 1(cid:19)
−
0 1 1 0 1 1
, .
(cid:18)1(cid:19)⊗ √2 (cid:18)1(cid:19) (cid:18)1(cid:19)⊗ √2 (cid:18) 1(cid:19)
−
The Hamilton operator Kˆ is given by the 4 4 matrix
×
~ω1 ~ω2 ǫ 0
~ω2 ~ω1 0 ǫ 
K = −
ǫ 0 ~ω1 ~ω2
 − 
e  0 ǫ ~ω2 ~ω1
 − − 
with the four eigenvalues
k1(ω1,ω2,ǫ) = ~2(ω1 +ω2)2 +ǫ2,
−
p
k2(ω1,ω2,ǫ) = ~2(ω1 +ω2)2 +ǫ2,
p
k3(ω1,ω2,ǫ) = ~2(ω1 ω2)2 +ǫ2,
− −
p
k4(ω1,ω2,ǫ) = ~2(ω1 ω2)2 +ǫ2
−
p
and the corresponding unnormalized eigenvectors
ǫ ǫ
 ǫ   ǫ 
, ,
k1 ~(ω1 +ω2) k2 ~(ω1 +ω2)
 −   − 
k2 +~(ω1 +ω2) k1 +~(ω1 +ω2)
   
ǫ ǫ
 ǫ   ǫ 
− , − .
k3 ~(ω1 ω2) k4 ~(ω1 ω2)
 − −   − − 
k3 ~(ω1 ω2) k4 ~(ω1 ω2)
 − −   − − 
Thus for the Hamilton operator Hˆ we have energy level crossing [10,12] which
is due to the discrete symmetry of the Hamilton operator Hˆ. For the Hamilton
operator Kˆ we have no energy level crossing for ǫ > 0. The symmetry is broken.
For ǫ and fixed frequencies the eigenvalues for the two Hamilton operators
→ ∞
approach ǫ (twice) and ǫ (twice). The four eigenvectors are entangled for ǫ > 0.
−
Extensions to higher order spin systems are straightforward. An extension is to
study the Hamilton operators with triple spin interactions
Hˆ = ~ω1(σx I2 I2)+~ω2(I2 σy I2)+~ω3(I2 I2 σz)
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
+γ12(σx σy I2)+γ13(σx I2 σz)+γ23(I2 σy σz)
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
+ǫ(σ σ σ )
x y z
⊗ ⊗
and
Kˆ = ~ω1(σx I2 I2)+~ω2(I2 σy I2)+~ω3(I2 I2 σz)
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
+γ12(σx σy I2)+γ13(σx I2 σz)+γ23(I2 σy σz)
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
+ǫ(σ σ σ ).
z y x
⊗ ⊗
Triplespin interacting systems have beenstudied by several authors[13,14,15]. For
Hˆ we find the eight product states given by the eigenstates of σ , σ , σ . We also
x y z
have energy level crossing owing to the symmetry of the Hamilton operator Hˆ. For
the Hamilton operator Kˆ the symmetry is broken and no level crossing occurs. We
also find entangled states for this Hamilton operator. As an entanglement measure
the three-tangle can be used [11]. Also the permutations σ σ σ , σ σ σ
z x y y z x
⊗ ⊗ ⊗ ⊗
of the interacting term could be investigated.
The question discussed in the introduction could also be studied for Bose systems
with a Hamilton operator such as
Hˆ = α(b†b I)+β(I (b† +b))+γ(b†b (b† +b))
⊗ ⊗ ⊗
where I is the identity operator and denotes the tensor product.
⊗
In conclusion we have shown that swapping the terms in the interacting part of
Hamilton operators acting in a product Hilbert space breaks the symmetry and
thus the behaviour about entanglement and energy level crossing will change.
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