Calculation of Magnetic Properties by Generalized Spin Hamiltonian and Generation of Global Entanglement: Cr Trimer in molecule and on surface
Oleg V. Stepanyuk2, Oleg V. Farberovich1
1 Raymond and Bekerly Sackler Faculty of Exact Sciences, School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv 69978, Israel
2 Max Planck Institute of Microstructure Physics, Halle, Germany
Here we present the results of the first-principles calculations of the spin structure of trimer Cr3 with the use of a density-functional scheme allowing for the non-collinear spin configurations in [1]. Using the results of these calculations we determine the Heisenberg-Dirac-Van Vleck (HDVV) Hamiltonian with anisotropic exchange couplings parameters linking the Cr ions with predominant spin density. The energy pattern was found from the effective HDVV Hamiltonian acting in the restricted spin space of the Cr ions by the application of the irreducible tensor operators (ITO)  technique. Comparison of the energy pattern with that obtained with the anisotropic exchange models conventionally used for the analysis of this system and with the results of non-collinear spin structure calculations show that our complex investigations provides a good description of the pattern of the spin levels and spin structures of the nanomagnetic trimer Cr3. The results are discussed in the view of a general problem of spin frustration related to the orbital degeneracy of the antiferromagnetic ground state. 
PACS numbers:
I. INTRODUCTION
Information technologies provide very interesting challenges and an extremely wide playground in which scientists working in materials science, chemistry, physics and nano-fabrication technologies may find stimuli for novel ideas. Curiously, the nanometre scale is the molecular scale. So we may wonder whether, how or simply which functional molecules can be regarded in some ways as possible components of nanodevices. The goal is ambitious:
it is not just a matter to store information in a 3dmetal trimer on a non-magnetic substrate, but we may think to process information with a trimer and then to communicate information at the supramolecules containg from magnetic 3d-metal trimer on a surface.
Spins are alternative complementary to charges as degrees of freedom to encode information. Recent examples, like for instance the discovery and application of Giant magnetoresistance in Spintronics, have demonstrated the efficient use of spins for information technologies.
Moreover, spins are intrinsically quantum entities and they have therefore been widely investigated in the field of quantum-information processing. Molecular nanomagnets are real examples of finite spin chains (1D) or clusters (0D), and therefore they constitute a new benchmark for testing models of interacting quantum objects.
New physics of molecular magnets feeds hopes of certain prospective applications, and such hopes pose the problem of understanding, improving, or predicting desirable characteristics of these materials. The applications which come into discussion are, for instance, magnetic storage (one molecule would store one bit, with much higher information storage density than accessible with microdomains of present-day storage media or magnetic nanoparticles of next future). In order to exploit the quantum features for information processing, molecular spin clusters have to fulfil some basic requirements.
Magnetic transition metal nanostructures on nonmagnetic substrates have attracted recently large attention due to their novel and unusual magnetic properties[2,3]. The supported clusters experience both the reduction of the local coordination number, as in free clusters, as well as the interactions with the electronic degrees of freedom of the substrate, as in embedded clusters. 
The complex magnetic behavior is usually associated with the competition of several interactions, such as interatomic exchange and bonding interactions, and in some cases noncollinear effects, which can give rise to several metastable states close in energy. The ground state can therefore be easily tuned by external action giving rise to the switching between different states.
In recent years, entanglement has attracted the attention of many physicists working in the area of quantum mechanics [1, 2]. This is due to the ongoing research in the area of quantum information [3]. Theoretical studies are also important in the context of spin interactions inside two structured reservoirs [9] such as single magnetic molecule (SMM) and metal cluster on nonmagnetic surface. Cr is unique among the transition-metal adatoms, because its half-filled valence configuration (3d54s1) yields both a large magnetic moment and strong interatomic bonding leading to magnetic frustration. We apply our method to Cr trimers deposited on a Au(111) surface and the trinuclear hydroxo-bridged chromium ammine complex [Cr3(NH3)10(OH)4]Br5 · 3H2O.
Low-lying excited states of a magnetic system are generally described in terms of a general spin-Hamiltonian.
For a magnetic system with many spin sites, this phenomenological Hamiltonian is written as a sum of pairwise spin exchange interactions between adjacent spin sites in molecule and surface.
In the present work we study entanglement between the spin states in the spin spectrum. In our model, a spin state interact with a continuum of the spin structure at interval temperature 0 – 300 K, and entanglement properties between the spin states in spin structure are considered. Using global entanglement as a measure of entanglement, we derive a pair of distributions that can be interpreted as densities of entanglement in terms of all the eigenvalue of the spin spectrum. This distribution can be calculated in terms of the spectrum of spin excitation of cluster surface and supramolecule. With these new measures of entanglement we can study in detail entanglement between the spin modes in spin structure.
The method developed here, in terms of entanglement distributions, can also be used when considering various types of structured reservoirs [..].
II. THE THEORETICAL APPROACH
In order to give a theoretical description of magnetic dimer we exploit the irreducible tensor operator (ITO) technique [ITO]. Let us consider a spin cluster of arbitrary topology formed from an arbitrary number of magnetic sites, N, with local spins S1, S2,…, SN which, in general, can have different values. A successive spin coupling scheme is adopted:
S1 + S2 = S˜2, S˜2 + S3 = S˜3, …, SN˜ô€€€1 + SN = S,
where ˜ S represents the complete set of intermediate spin quantum numbers S˜k, with k=1,2,…,N-1.The eigenstates | v⟩ of spin-Hamiltonian will be given by linear combinations of the basis states | ( ˜ S)SM⟩:
| v⟩ =
Σ
(~S
)SM
⟨(~S
)SM | v⟩ | (~S
)SM⟩, (1)
where the coefficients ⟨( ˜ S)SM | v⟩ can be evaluated once the spin-Hamiltonian of the system has been diagonalized.
Since each term of spin-Hamiltonian can be rewritten as a combination of the irreducible tensor operators technique[ITO].In [ITO] work focus on the main physical interactions which determine the spin-Hamiltonian and to rewrite them in terms of the ITO’s. The exchange part of the spin-Hamiltonian is to introduced:
Hspin = H0 + HBQ + HAS + HAN. (2)
The first term H0 is the Heisenberg-Dirac Hamiltonian, which represents the isotropic exchange interaction, HBQ is the biquadratic exchange Hamiltonian, HAS is the antisymmetric exchange Hamiltonian,and HAN represents the anisotropic exchange interaction. Conventionally, they can be expressed as follows [ITO]:
H0 = −2
Σ
i;f
Jif bSi bSf (3)
HBQ = −
Σ
i;f
jif ( bSi bSf )2 (4)
HAS =
Σ
i;f
Gif [ bSi × bSf ] (5)
HAN = −2
Σ
i;f
Σ
_
J_
if
bS_
i
bS_
f (6)
with α = x, y, z We can add to the exchange Hamiltonian the term due to the axial single-ion anisotropy:
HZF =
Σ
i
Di bSz(i)2 (7)
where Jif and J_ 
if are the parameters of the isotropic and anisotropic exchange iterations, jif are the coefficients of the biquadratic exchange iterations,and Gif=-Gfi is the vector of the antisymmetric exchange. The terms of the spin-Hamiltonian above can be written in terms of the ITO’s.
Both the Heisenberg–Dirac and biquadratic exchange are isotropic interactions. In fact, the corresponding Hamiltonians can be described by rank-0 tensor operators and thus have non zero matrix elements only with states with the same total spin quantum number S (ΔS,ΔM=0). The representative matrix can be decomposed into blocks depending only on the value of S and M. All anisotropic terms are described by rank-2 tensor operators which have non zero matrix elements between state with ΔS=0,±1,±2 and their matrices can not be decomposed into blocks depending only on total spin S in account of the S–mixing between spin states with different S. The single-ion anisotropy can be written in terms of rank-2 single site ITO’s [ITO]. Finally, the antisymmetric exchange term is the sum of ITO’s of rank-1.
The ITO technique has been used to design the MAGPACK software [ITO1], a package to calculate the energy levels, bulk magnetic properties, and inelastic neutron scattering spectra of high nuclearity spin clusters that allows studying efficiently properties of nanoscopic magnets.
A. Calculation of the magnetic properties
Once we have the energy levels, we can evaluate different thermodynamic properties of the system as magnetization, magnetic susceptibility, and magnetic specific heat. Because anisotropic interactions are not included, the magnetic properties of the anisotropic system do not depend on the direction of the magnetic field. For this reason one can consider the magnetic field directed along arbitrary axis Z of the molecular coordinate frame that is chosen as a spin quantization axis. In this case the energies of the system will be ϵ_(Ms)+geβMsHZ, where Ïµ_(Ms) are the eigenvalues of the Hamiltonian containing magnetic exchange and double exchange contributions (index μ runs over the energy levels with given total spin protection Ms). Then the partition function in the presence of the external magnetic field is given by:
Z(HZ) =
Σ
Ms;_
exp[−ϵ_(Ms)/kT]
Σ
Ms
exp[−geβMsHZ/kT]
(8)
Using this expression one can evaluate the magnetic susceptibility χ and magnetization M by standart thermodinamical definitions:
χ =
(
∂M
∂H
)
H!0
(9)
M(H) = NkT
∂lnZ
∂H
(10)
B. Entanglement in N-spin system
Entanglement has gained renewed interest with the development of quantum information science. The problem of measuring entanglement is a vast and lively field of research in its own. In this section we attempt to solve the problem of measuring entanglement in the N-spin cluster and supramolecules systems. Based on the residual entanglement [9] (Phys. Rev. A 71, 044301 (2005)), we present the global entanglement for a N-spin state for the collective measures of multiparticle entanglement. This measures introduced by Meyer andWallach[..]. The MeyerWallach (MW) measure written in the Brennen form
(G.K.Brennen,Quantum.Inf.Comp.,v.3,619 (2003)) is:
Q(ψ) = 2(1 − 1
N
ΣN
k=1
Tr[ρ2
k]) (11)
where ρk is the reduced density matrix for k-th qubit. 
The problem of entanglement between a spin states in N-spin systems is becoming more interesting when considering clusters or molecules with a spectral gap in their densities of states. For quantifying the distribution of entanglement between the individual spin eigenvalues in spin structure of N-spin system we use the density of entanglement.
The density of entanglement ε(ϵ_, ϵ_, ϵ)dϵ
gives the entanglement between the spin eigenvalue ϵ_
and spin eigenvalue ϵ_ with in an energy interval ϵ_ to
ϵ_ + dϵ_.
One can show that entanglement distribution can be written in terms of spectrum of spin exitation
S(ϵ_, ϵ) = |c_|2 δ(ϵ − ϵ_) (12)
and
ε(ϵ_, ϵ_, ϵ) = 2S(ϵ_, ϵ)S(ϵ_, ϵ) (13)
where coefficient c_ = ⟨( ˜ S)SM | v⟩ is eigenvector of the spin-Hamiltonian of the cluster or supramolecule. Thus, entanglement distributions can be derived from the excitation spin spectrum
Q(ϵ) = 1− 2Δ2
π2N
ΣN
_=1
|c_|2
(ϵ − ϵ_)2 + Δ2
ΣN
_=_+1
|c_|2
(ϵ − ϵ_)2 + Δ2
(14)
Though the very nature of entanglement is purely quantum mechanical, we saw that it can persist for macroscopic systems and will survive even in the thermodynamical limit. In this section we discuss how it behaves at finite temperature of thermal entanglement.
The states in N-spin system describing a system in thermal equilibrium states, are determined by the Generalized spin-Hamiltonian and thermal density matrix
ρ(T) =
exp(−Hspin/kT)
Z(HZ)
(15)
where Z(HZ) is the partition function of the N-spin system.
The thermal entanglement is
Q(ϵ, T,HZ) = 1 − 2Δ2
π2NZ(HZ)2
ΣN
_=1
|c_|2 exp[−ϵ_/kT]
(ϵ − ϵ_)2 + Δ2
×
(16)
ΣN
_=_+1
|c_|2 exp[−ϵ_/kT]
(ϵ − ϵ_)2 + Δ2
The demonstration of quantum entanglement, however, can also be directly derived from experiments, without requiring knowledge of the system state. This can be done by using specific operators–the so-called entanglement witnesses–whose expectation value is always positive if the state ρ is factorizable. It is quite remarkable that some of these entanglement witnesses coincide with well-known magnetic observables, such as energy or magnetic susceptibility χ = dM/dB. In particular, the magnetic susceptibility of N spins s, averaged over three orthogonal spatial directions, is always larger than a threshold value if their equilibrium state ρ is factorizable:
Σ
g χg > Ns/kBT [EW]. This should not be surprising, since magnetic susceptibility is proportional to the variance of the magnetization, and thus it may actually quantify spin.spin correlation. The advantage in the use of this criterion consists in the fact that it does not require knowledge of the system Hamiltonian, provided that this commutes with the Zeeman terms corresponding to the three orthogonal orientations of the magnetic field α = x, y, z. As already mentioned, in the case of the Cr3 trimer, the effective Hamiltonian includes, besides the dominant Heisenberg interaction J ∼118 meV , smaller anisotropic terms G ∼ 1.1 meV and D ∼ 0.18 meV , due to which the above commutation relations are not fulfilled. This might, in principle, result in differences between the magnetic susceptibility and the entanglement witness WE (see Fig.). Apparently, the difference is quite essential and therefore it is necessary to use a formula for global entanglement Q(ψ) in N-spin system.
4
10−1 100 101 102 103
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
The calculated difference EW(T)−EWa(T)/EW(T)for Cr3 isosceles trimer
T(K)
EW(T)−EWa(T)/EW(T)
FIG. 1: (Color online) The calculated difference j EW(T) ô€€€
EWa(T) j =EW(T) for Cr3 isosceles trimer
0
100
200
300
400
0
2
4
6
0
0.2
0.4
0.6
0.8
1
1.2
Angle(Degrees)
The calculated M(H) for Cr3 isosceles trimer
H(T)
M(B)
FIG. 2: (Color online)Magnetization M(H) of the Cr3 isoscales trimer on metal surface as a function of angles from 0 to 360 degree
C. Thermal global entanglement in static magnetic
_eld
5
0 50 100 150 200 250 300 350 400
0
0.05
0.1
0.15
0.2
0.25
The calculated variation of M(H) vs angle (magnetization switching)
Angle(Degrees)
M(B)
0.1Ts
0.2Ts
0.5Ts
1.0Ts
FIG. 3: (Color online)The calculated variation of M(H) vs angle (magnetization switching) for Cr3 isoscales trimer 
FIG. 4: (Color online)The calculated density of global entanglement vs temperature and energy for Cr3 isoscales trimer
6
0
100
200
300
400
0
2
4
6
0
0.5
1
1.5
2
2.5
Angle(Degrees)
The calculated M(H) for Cr3 molecular magnet
H(T)
M(B)
FIG. 5: (Color online)Magnetization M(H) of the Cr3 molecular magnet as a function of angles from 0 to 360 degree
0 50 100 150 200 250 300 350 400
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
The calculated variation of M(H) vs angle (magnetization switching)
Angle(Degrees)
M(B)
0.1Ts
0.2Ts
0.5Ts
1.0Ts
FIG. 6: (Color online)The calculated variation of M(H) vs angle (magnetization switching) for Cr3 molecular magnet
7
FIG. 7: (Color online)The calculated density of global entanglement vs temperature and energy for Cr3 molecular magnet FIG. 8: (Color online)The calculated entanglement for the Cr3 isoscales trimer as a function of temperature and the magnitude of the magnetic field Hpar.
8
FIG. 9: (Color online)The calculated entanglement for the Cr3 isoscales trimer as a function of temperature and the magnitude of the magnetic field Hper.
FIG. 10: (Color online)The calculated entanglement for the Cr3 isoscales trimer as a function of temperature and the magnitude of the magnetic field Hav.
9
FIG. 11: (Color online)The calculated entanglement for the Cr3 molecular magnet as a function of temperature and the magnitude of the magnetic field Hav.