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

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

Willi-Hans Steeb^a, Yorick Hardy^b, and Jacqueline de Greef^a

aInternational School for Scientific Computing, University of Johannesburg, Auckland Park 2006, South Africa

bDepartment of Mathematical Sciences, University of South Africa, Pretoria, South Africa Reprint requests to W.-H. S.; E-mail:steebwilli@gmail.com

Z. Naturforsch.67a,608 – 612 (2012) / DOI: 10.5560/ZNA.2012-0064 Received June 12, 2012 / published online August 20, 2012

We study finite-dimensional product Hilbert spaces, coupled spin systems, entanglement, and energy level crossing. The Hamilton operators are based on the Pauli group. We show that swapping the interacting term can lead from unentangled eigenstates to entangled eigenstates and from an energy spectrum with energy level crossing to avoided energy level crossing.

Key words:Hilbert Space; Energy Level Crossing; Discrete Symmetries; Entanglement.

1. Introduction

LetH₁,H₂ be Hilbert spaces andH₁⊗ H₂be the tensor product Hilbert space [1,2]. Quite often a self- adjoint Hamilton operator acting on the tensor product Hilbert spaceH₁⊗ H₂can be written as

Hˆ =Hˆ₁⊗I₂+I₁⊗Hˆ₂+εVˆ, (1) where the self-adjoint Hamilton operator ˆH₁acts in the Hilbert space H₁, the self-adjoint Hamilton operator Hˆ₂acts in the Hilbert spaceH₂,I₁is the identity operator acting in the Hilbert spaceH₁, andI₂is the identity operator acting in the Hilbert spaceH₂. The self- adjoint operator ˆVacts in the product Hilbert space and ε is a real parameter. The main task would be to find the spectrum of ˆH.

In the following we consider the finite-dimensional Hilbert spaceH₁=H₂=Cⁿand then⊗denotes the Kronecker product [3–6]. LetI_nbe then×nidentity matrix. We consider the two hermitian Hamilton operators

Hˆ =αA⊗I_n+In⊗βB+ε(A⊗B), (2) Kˆ=αA⊗I_n+I_n⊗βB+ε(B⊗A), (3) whereA,Bare nonzeron×nhermitian matrices and α, β,ε are real parameters with ε≥0. We assume that[A,B]6=0. The vector space of then×nmatrices

over Cform a Hilbert space with the scalar product hX,Yi:=tr(XY^∗). We also assume thathA,Bi=0, i.e.

the nonzeron×n hermitian matricesAandBare or- thogonal to each other. Of particular interest would be the case whereAandBare 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 ofAandB. In the following the matricesAandBare realized by Pauli spin matrices. The Hamilton operator will be a linear combination of elements of the Pauli groupP_n. The Pauli group [7] is defined by

P_n:={I₂,σx,σy,σz}^⊗n⊗ { ±1,±i}. (4) Such two-level and higher level quantum systems and their physical realization have been studied by many authors (see [8] and references therein). The thermo- dynamic behaviour is determined by the partition functions

Z_H_ˆ(β) =tr(exp(−βH)),ˆ Z_K_ˆ(β) =tr(exp(−βK)).ˆ Since

tr(H) =ˆ tr(Kˆ)

=αntr(A) +βntr(B) +ε(tr(A))(tr(B)), the sum of the eigenvalues of the operators ˆHand ˆKare the same. However, in general, the partition functions will be different.

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2. Commutators, Eigenvalues, and Eigenvectors Let us first summarize the equations we utilize in the following. LetA,Bben×nmatrices overC. First note that we have the following commutators:

[A⊗I_n,I_n⊗B] =0, [A⊗I_n,A⊗B] =0, [I_n⊗B,A⊗B] =0 and

[A⊗I_n,B⊗A] = ([A,B])⊗A, [I_n⊗B,B⊗A] =B⊗([B,A]).

The last two commutators would be 0 if [A,B] =0.

There is ann²×n²permutation matrixP(swap gate) such that P(A⊗B)P⁻¹=B⊗A. This implies that P(A⊗I_n)P⁻¹=I_n⊗AandP(I_n⊗B)P⁻¹=B⊗I_n.

Now letAandBben×nhermitian matrices. If the eigenvalues and normalized eigenvectors of A and B areλ_j,u_j,µ_j,v_j,(j=1,2, . . . ,n), respectively, then the eigenvalues and normalized eigenvectors of the Hamil- ton operator (2) are given by [3–6]

α λ_j+β µ_k+ε λ_jµ_k, u_j⊗v_k j,k=1,2, . . . ,n. Thus the eigenvectors are not entangled since they can be written as product states. These results can be ex- tended to the Hamilton operator

Hˆ =α(A⊗I_n⊗I_n) +β(I_n⊗B⊗I_n) +γ(I_n⊗I_n⊗C) +ε(A⊗B⊗C)

and higher dimensions.

3. Pauli Spin Matrices and Entanglement

Since we realize the linear operators A andB by Pauli spin matrices we summarize some results for the Pauli spin matrices and their Kronecker products. Con- sider the Pauli spin matricesσ_z,σ_x,σ_y. The eigenvalues are given by +1 and−1 with the corresponding normalized eigenvectors

1 0

, 0

1

, 1

√2 1

1

, 1

√2 1

−1

,

√1 2

−i 1

, 1

√ 2

i 1

forσz,σx, andσy, respectively. Consider now the three hermitian and unitary 4×4 matricesσx⊗σx,σy⊗σy, σ_z⊗σ_z. These matrices appear in Mermin’s magic 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 matricesσx⊗σx,σy⊗σy,σz⊗σz are+1 (twice) and

−1 (twice). The eigenvectors can be given as product states (unentangled states), but also as entangled states (i.e. they cannot be written as product states). Obvi- ously,

1 0

⊗ 1

0

, 1

0

⊗ 0

1

, 0

1

⊗ 1

0

, 0

1

⊗ 0

1

are four normalized product eigenstates ofσz⊗σ_z. The normalized product eigenstates ofσx⊗σxare

1 2

1 1

⊗ 1

1

, 1

2 1

1

⊗ 1

−1

, 1

2 1

−1

⊗ 1

1

, 1

2 1

−1

⊗ 1

−1

.

The normalized product eigenstates ofσy⊗σyare 1

2 i

1

⊗ i

1

, 1

2 i

1

⊗ −i

1

, 1

2 −i

1

⊗ i

1

, 1

2 −i

1

⊗ −i

1

.

All three 4×4 matrices also admit the Bell basis

√1 2





 1 0 0 1







, 1

√2





 0 1 1 0







, 1

√2





 1 0 0

−1







, 1

√2





 0 1

−1 0







as normalized eigenvectors which are maximally entangled. As measure for entanglement the tangle [5,7, 10,11] will be utilized.

Consider now the hermitian and unitary 4×4 matrices σx⊗σz, σz⊗σx. Since the eigenvalues of the Pauli matrices are given by+1 and−1, the eigenvalues of the 4×4 matricesσx⊗σzandσz⊗σx, are+1 (twice) and−1 (twice). The eigenvectors can be given as product states (unentangled states), but also as entangled states (i.e. they cannot be written as product states).

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The normalized product eigenstates ofσx⊗σzare

√1 2

1 1

⊗ 1

0

, 1

√2 1

1

⊗ 0

1

,

√1 2

1

−1

⊗ 1

0

, 1

√ 2

1

−1

⊗ 0

1

.

The normalized product eigenstates ofσz⊗σxare

√1 2

1 0

⊗ 1

1

, 1

√2 1

0

⊗ 1

−1

,

√1 2

0 1

⊗ 1

1

, 1

√ 2

0 1

⊗ 1

−1

.

The two 4×4 matrices also admit

1 2







−1

−1 1







, 1

2







−1 1 1 1







, 1

2





 1

−1 1 1







, 1

2





 1 1

−1 1







as normalized eigenvectors which are maximally entangled. Note thatσ_y⊗σ_yalso admits these maximally entangled eigenvectors besides the Bell basis as eigenvectors and the product eigenvectors.

For the triple spin interaction termσx⊗σy⊗σz, we obtain the eigenvalues+1 (fourfold) and−1 (fourfold) and all the eight product states as eigenstates given by the eigenstates ofσx,σy,σz. Owing to the degeneracies of the eigenvalues, we also find fully entangled states such as

1

2 1 1 0 0 0 0 i −iT

with the three-tangle as measure [11].

4. Examples

Consider now a specific example for αA and βB withn=2 andε>0. Utilizing the Pauli spin matrices

αA=hω¯ ₁σz, βB=hω¯ ₂σx,

whereα=hω¯ ₁,β =hω¯ ₂andω₁,ω₂are the frequencies. Note that[σ_z,σ_x] =2iσ_yand tr(H) =ˆ 0, tr(K) =ˆ 0.

The elements of the set

{I2⊗I₂,σ_z⊗I2,I2⊗σ_x,σ_z⊗σ_x}

form a commutative subgroup of the Pauli groupP₂. The elementsσ_z⊗I₂,I₂⊗σ_x,σ_x⊗σ_zare generators of

the Pauli groupP₂. Now the eigenvalues and eigenvectors ofαAare given by

λ1=hω¯ 1, u₁= 1

0

, λ2=−¯hω1, u₂= 0

1

, and the eigenvalues and eigenvectors ofβBare given by

µ1=hω¯ ₂, u₁= 1

√ 2

1 1

,

µ2=−¯hω₂, u₂= 1

√2 1

−1

.

The Hamilton operator ˆHis given by the 4×4 matrix which can be written as direct sum of two 2×2 matrices:

He=







¯

hω₁ hω¯ ₂+ε 0 0

¯

hω₂+ε hω¯ ₁ 0 0

0 0 −¯hω₁ hω¯ ₂−ε

0 0 hω¯ 2−ε −¯hω1





 .

The eigenvalues ofHeare

E₁(ω₁,ω₂,ε) =hω¯ ₁+hω¯ ₂+ε, E₂(ω₁,ω2,ε) =hω¯ ₁−hω¯ ₂−ε, E₃(ω₁,ω2,ε) =−¯hω₁+hω¯ ₂−ε, E₄(ω₁,ω2,ε) =−¯hω₁−hω¯ ₂+ε

with the corresponding eigenvectors (which can be written as product states)

1 0

⊗ 1

√ 2

1 1

, 1

0

⊗ 1

√ 2

1

−1

, 0

1

⊗ 1

√2 1

1

, 0

1

⊗ 1

√2 1

−1

. The Hamilton operator ˆKis given by the 4×4 matrix

Ke=







hω¯ 1 hω¯ 2 ε 0

¯

hω₂ hω¯ ₁ 0 −ε ε 0 −¯hω₁ hω¯ ₂ 0 −ε hω¯ ₂ −¯hω₁







with the four eigenvalues k₁(ω₁,ω₂,ε) =−

q

¯

h²(ω₁+ω₂)²+ε²,

k₂(ω₁,ω₂,ε) = q

¯

h²(ω₁+ω₂)²+ε²,

k3(ω₁,ω2,ε) =− q

¯

h²(ω₁−ω2)²+ε², k4(ω₁,ω2,ε) =

q

¯

h²(ω₁−ω2)²+ε²

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and the corresponding unnormalized eigenvectors







ε ε k₁−h(ω¯ ₁+ω2) k2+h(ω¯ ₁+ω2)





 ,







ε ε k₂−h(ω¯ ₁+ω₂) k1+h(ω¯ ₁+ω₂)





 ,







ε

−ε k₃−h(ω¯ ₁−ω2) k₃−h(ω¯ ₁−ω₂)





 ,







ε

−ε k₄−h(ω¯ ₁−ω2) k₄−h(ω¯ ₁−ω₂)





 .

Thus for the Hamilton operator ˆH, we have energy level crossing [10,12] which is due to the discrete symmetry of the Hamilton operator ˆH. The permuta- tion matricesPwithPHPˆ ^T=Hˆ are given byP0=I4

and

P₁=







1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0





 , P₂=







0 1 0 0

1 0 0 0

0 0 1 0

0 0 0 1





 ,

P₃=







0 1 0 0

1 0 0 0

0 0 0 1

0 0 1 0





 ,

whereP0=I4is the 4×4 identity matrix. The matri- cesP₀,P₁,P₂,P₃form a commutative group under matrix multiplication. These permutation matrices satisfy P_j²=I₄for j=1,2,3,4. Thus ¹₂(I₄+P_k)and¹₂(I₄−P_k) (k=1,2,3) are projection matrices and the Hilbert spaceC⁴can be decomposed into invariant sub Hilbert spaces. In the present caseC²andC².

For the Hamilton operator ˆK, we have no energy level crossing forε>0. The symmetry is broken, i.e.

the Hamilton operator ˆK only admitsP₀=I₄as discrete symmetry. Forε→∞and fixed frequencies, the eigenvalues for the two Hamilton operators approachε (twice) and−ε(twice). The four eigenvectors are entangled forε>0.

Extensions to higher order spin systems such as spin-1 are straightforward. An extension is to study the

Hamilton operators with triple spin interactions:

Hˆ=hω¯ ₁(σ_x⊗I₂⊗I₂) +hω¯ ₂(I₂⊗σ_y⊗I₂) +hω¯ ₃(I₂⊗I₂⊗σ_z) +γ₁₂(σ_x⊗σ_y⊗I₂) +γ₁₃(σ_x⊗I₂⊗σ_z) +γ₂₃(I₂⊗σ_y⊗σ_z) +ε(σ_x⊗σ_y⊗σ_z)

and

Kˆ=hω¯ ₁(σ_x⊗I₂⊗I₂) +hω¯ ₂(I₂⊗σy⊗I₂) +hω¯ ₃(I₂⊗I₂⊗σz) +γ12(σ_x⊗σy⊗I₂) +γ13(σ_x⊗I₂⊗σz) +γ23(I₂⊗σy⊗σz) +ε(σ_z⊗σ_y⊗σx).

Triple spin interacting systems have been studied by several authors [13–15]. For ˆH, we find the eight prod- uct states given by the eigenstates ofσx,σy,σz. We also have energy level crossing owing to the symmetry of the Hamilton operator ˆH. For the Hamilton opera- tor ˆK, 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 permutationsσz⊗σx⊗σy, σ_y⊗σ_z⊗σ_x of the interacting term could be investi- gated.

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)), whereIis 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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