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Physics Letters A
www.elsevier.com/locate/pla
Quantification of entanglement from magnetic susceptibility for a Heisenbergspin 1/2 system
Tanmoy Chakraborty, Harkirat Singh, Diptaranjan Das, Tamal K. Sen, Chiranjib Mitra ∗
Indian Institute of Science Education and Research (IISER) Kolkata, Mohanpur Campus, PO BCKV Campus Main Office, Mohanpur – 741252, Nadia, West Bengal, India
a r t i c l e i n f o a b s t r a c t
Article history:Received 1 May 2012Received in revised form 19 July 2012Accepted 29 August 2012Available online 1 September 2012Communicated by P.R. Holland
Keywords:Spin chainAntiferromagnetismEntanglement
We report temperature and magnetic field dependent magnetization and quantification of entanglementfrom the experimental data for dichloro (thiazole) copper (II), a Heisenberg spin chain system. Theplot of magnetic susceptibility vs. temperature indicates an infinite spin chain. Isothermal magnetiza-tion measurements (as functions of magnetic field) were performed at various temperatures below theantiferromagnetic (AFM) ordering, where the AFM correlations persist significantly. These magnetizationcurves are fitted to the Bonner–Fisher model. Magnetic susceptibility is used as an entanglement witnessto quantify the amount of entanglement in the system.
The study of entanglement in quantum systems is an area ofintense research primarily due to its usefulness as a fundamentalresource in quantum computing and quantum communication [1].Idea of entanglement lies in the fundamentals of quantum me-chanics and it was first discussed by Einstein in 1935 [2]. Entan-glement refers to nonlocal quantum correlations between differentquantum systems that are entirely nonclassical in nature and hardto visualize within the purview of the classical world. In the pastcouple of decades it has been studied extensively both theoreti-cally and experimentally from the context of quantum informationscience. For instance, it has important applications in the areaof quantum information processing [3] and quantum cryptogra-phy [4]. Hence, the study of entanglement in condensed matter isrelevant while dealing with many body systems having quantumcorrelations between its constituent particles [5].
Measures of entanglement in many body condensed matter sys-tem is an area of active research. It has been reported that entan-glement is quantifiable and it is possible to measure the quantityof entanglement in a solid state system by using EntanglementWitnesses (EW) [6–11]. EW is an observable that takes up nega-tive values for separable states and positive values for entangledstates. Quantification of entanglement from experimental data wasfirst experimentally done by carrying out bulk measurement on aninsulating magnetic system LiHoxY(1−x)F [12]. Consequently, vari-
ous measures of entanglement have been proposed. In their work,Wiesniak et al. [6] have discussed how entanglement in certainsystems can have signatures even in the thermodynamic limit. Ithas already been reported that macroscopic thermodynamic ob-servables like susceptibility and specific heat can capture nonlocalcorrelations amongst the microscopic constituents (spins) in solidstate systems. This could be used to detect and quantify entangle-ment in these systems using the fact that non-separability leads toviolation of certain uncertainty bounds which were obtained as-suming separability of constituent subsystems [6,13]. In this work,magnetic susceptibility is used as an entanglement witness andquantification of entanglement is done by extracting informationfrom the magnetic susceptibility data.
One dimensional exchange coupled magnetic systems havebeen studied extensively both theoretically and experimentally.However, studying different spin chain systems from the perspec-tive of quantum information science has been a field of greatinterest in recent times. A prospective application of spin 1/2chains, where the spins are entangled, is that they can be usedas a medium for quantum communication [14,15]. Spin 1/2 chainsystems provide an excellent platform for exploring the entangle-ment content in a many body system. This is because, due to thesymmetry in the Hamiltonian of these systems, exact numerical di-agonalization is possible [16]. The variation of entanglement withmagnetic field and temperature has been shown experimentally fora spin 1/2 system with Heisenberg interaction [17,18]. In absenceof an external field the ground state of an antiferromagnetic sys-tem is entangled, because the order parameter (which is staggeredmagnetization for systems with J > 0) does not commute with theHamiltonian, leading to quantum fluctuations which prevent long
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range ordering. However, for the ferromagnetic case the groundstate is not entangled. This happens because there is no spin fluc-tuation at low temperature as the order parameter commutes withthe Hamiltonian [18].
We report detailed magnetic study and quantification of en-tanglement from experimentally obtained magnetic susceptibilitydata for dichloro (thiazole) copper (II), a Cu based polymer sys-tem [19,20]. It has been confirmed from energy calculations thatantiferromagnetic state is a stable ground state for this system[19]. Magnetic centre for this system, resides in the Cu atom andeach molecule bears one unpaired spin. The structure consists ofsquare-planer units arranged in arrays. These units are equatoriallylinked along a particular axis by chloride bridges to form infinitelinear chains [19,20]. Earlier study has shown [20] that for this sys-tem, unpaired spins interact with other neighbouring spins alonga particular direction by isotropic Heisenberg interaction. This pe-riodic arrangement of spins along a particular direction enables usto represent this system like a spin 1/2 chain that is translation-ally invariant. It has been shown [20] that the magnetic interactionin this compound is one dimensional by fitting its susceptibilityvs. temperature data to Bonner–Fisher model in low temperatureregime. It has, however, small interchain interaction at relativelyhigher temperatures. To study entanglement in this system, thelow temperature regime is important and hence one can considerthis as a 1D system. The intrachain interaction between the nearestneighbours is antiferromagnetic with the same coupling strength.Bonner and Fisher discussed different cases of an interacting spinchain Hamiltonian. The general form of the spin chain Hamiltonianfor a finite number of spins with nearest neighbour interaction is
H = 2 J∑
i
[αSz
i Szi+1 + β
(Sx
i Sxi+1 + S y
i S yi+1
)] + B∑
i
S zi (1)
where 0 � α � 1 and 0 � β � 1. Here, J is the exchange integral,B is the external magnetic field and Sx , S y , Sz are the componentsof the total spin S along x, y and z directions respectively. WhenJ > 0, the system becomes antiferromagnetic. For α = 1 and β = 0,
Ising model is obtained which represents the fully classical regimeand when α = β = 1, the Hamiltonian takes the form of isotropicHeisenberg model which is the fully quantum mechanical regime.Comparison between these two scenarios is shown graphically inFig. 1(a). For the Ising case, the preferred direction of alignment ofthe spins is the z-axis. However, for the isotropic Heisenberg case,there is no preferred direction of alignment of the spins, whichmakes it rotationally invariant. The spin system that we have stud-ied here is an isotropic Heisenberg system and the Hamiltoniancan be best described by Eq. (1) with α = β = 1. Bonner and Fisher[21] carried out calculations on isotropically interacting spin chainsystems varying the number of spins from 3 to 11 and extrapolatedit for infinite chain with a good agreement. Hall [22] found an ex-cellent analogy of the numerical Bonner–Fisher model for zero fieldsusceptibility to the expression given below (Eq. (2)) and fitted itefficiently to the theoretically calculated Bonner–Fisher data,
χ ≈ Ng2μ2B
kB T· 0.25 + 0.14995X + 0.30094X2
1.0 + 1.9862X + 0.68854X2 + 6.0626X3(2)
Here, X = J/kB T , T is temperature, kB is the Boltzmann constant,g is the Landé g-factor, N is the Avogadro’s number and μB isthe Bohr magneton. Magnetic susceptibility is experimentally mea-sured as a function of temperature and fitted to the above analyti-cal expression. We obtained a good fit consistent with the previousliterature [20]. Subsequently, magnetization isotherms with varyingmagnetic field are taken at different temperatures. These exper-imentally measured magnetic datasets are used to quantify theamount of entanglement in this spin chain system.
Fig. 1. (a) Susceptibility data (circles) of Cu(tz)2Cl2 and fit (solid line) to Eq. (2), theisotropic Heisenberg chain model. The dotted line demarks the entangled regime(see text). The solid blue line represents the Ising model. (b) Quantification of en-tanglement witness from the susceptibility data of Cu(tz)2Cl2 using Eq. (3).
2. Materials and methods
Thiazole (C3H3NS), Copper chloride dyhydrate (CuCl2, 2H2O)and absolute Ethanol of purest grade, obtained from SIGMAALDRICH, were used as starting reagents. We followed the synthe-sis route as mentioned in Ref. [20] and obtained nice blue crystals.The magnetic measurements of Cu(tz)2Cl2 were performed in aQuantum Design MPMS (Magnetic Property Measurement System).The static magnetic susceptibility vs. temperature data were col-lected in a temperature range of 2 K to 24 K. Subsequently, mag-netization isotherms as a function of magnetic field were takenat temperatures ranging from 2 K to 20 K. The field was variedfrom 0 to 7 Tesla.
3. Results and discussion
We have used magnetic susceptibility as a macroscopic entan-glement witness to analyze the results of magnetic measurementsfor a spin 1/2 antiferromagnetic chain. Experimentally measuredmagnetic susceptibility as a function of temperature is shown inFig. 1(a). The data is fitted to Bonner–Fisher model using the an-alytical expression given in Eq. (2) while taking into account theboundary effects, which results paramagnetic contribution to thesusceptibility data. The fitting was done using the method of non-linear curve fitting. The contribution of paramagnetic impurity tothe total susceptibility is 1.35% [20] and this is subtracted from theexperimental data. Exchange coupling constant J was used as onlyfitting parameter and g was taken 2.06 [20]. The value of J thatwas obtained after fitting is 5.6 K, which is in good agreement with
T. Chakraborty et al. / Physics Letters A 376 (2012) 2967–2971 2969
Fig. 2. 3D plot showing magnetization, magnetic field and temperature along the three axes for Cu(tz)2Cl2.
the value reported in the previous literature [20]. An EntanglementWitness (EW) is able to certify if a certain system is entangled ornot and it was first introduced by Horodecki et al. [23]. An ob-servable W corresponding to a state ρ can be used as an EW ifTr(ρW ) < 0, when ρ is a separable state. When ρ is entangledTr(ρW ) � 0 [24]. Magnetic susceptibility is a macroscopic entan-glement witness, which for spin 1/2 chain consisting of N spins isgiven as
EW(N) =[
1 − 6kB Tχ
g2μ2B N
](3)
Here χ is magnetic susceptibility that can be measured experi-mentally and the other symbols have their usual meanings. EWprovides necessary and sufficient condition for existence of en-tanglement. For a state to be entangled, the expectation value ofEW at that particular state should be within a certain bound; al-though expectation value exceeding the bound does not assureseparability [5]. In our case, the condition 1 >
6kB Tχ
g2μ2B N
implies exis-
tence of entanglement in the system. The entanglement boundaryis shown by the dotted curve (Fig. 1(a)). The entangled region isrepresented by the region towards the left of the dotted curve andthis demarcation is governed by the condition “1 >
6kB Tχ
g2μ2B N
”. This
entanglement boundary intersects the susceptibility curve at 12 K.That means the system exhibits entanglement up to 12 K. This isthe critical value of the temperature above which the system mayor may not remain entangled. The entanglement exists up to atemperature (12 K in this case) where the antiferromagnetic corre-lations persist and this value of temperature is more than the mag-netic ordering temperature. The magnetic ordering temperaturecorresponds to the peak in the susceptibility curve, which is 7.3 Kin this case. The magnetic ordering in this case is not strictly a longrange ordering akin to a classical magnetic system. The presence ofspin fluctuations mentioned above, prevents any kind of long rangeorder. The value of EW is estimated from experimentally measuredtemperature dependent susceptibility data using the mathematicalexpression for EW (Eq. (3)). The quantified value of EW is plot-ted as a function of temperature and is shown in Fig. 1(b). Sincewe are only concerned with values of EW in the region below thecritical temperature, all EW values above this particular tempera-ture are considered to be zero. One can see that EW vanishes at12 K which is the critical temperature mentioned above.
Fig. 3. Experimental data of magnetization collected at T = 3 K and 5 K forCu(tz)2Cl2 (open circles and open squares respectively) as a function of magneticfield. Solid red line and solid blue line represent the theoretical curves at T = 3 Kand 5 K derived using the Bonner–Fisher model. (For interpretation of the refer-ences to colour in this figure legend, the reader is referred to the web version ofthis Letter.)
Isothermal magnetization measurements as a function of mag-netic field for various temperatures were performed for Cu(tz)2Cl2.These measurements are concentrated in the temperature regimewhere magnetic ordering occurs, that is 7.3 K in this case. Magne-tization isotherms are taken in the temperature range 3 K to 24 K.A 3D plot (Fig. 2) is generated from experimental data that explic-itly shows the variation of magnetization with magnetic field andtemperature for this system.
The magnetization isotherms at 3 K and 5 K are compared withthe theoretical Bonner–Fisher model of spin chain for 10 spins.Bonner–Fisher numerically generated isothermal magnetization for10 isotropically interacting spins and assured the convergencefor N tending to infinity. We have numerically calculated theoreti-cal magnetization for 10 spins as a function of field and plotted theresults along with the experimental ones at corresponding temper-atures. We have used J = 5.6 K (obtained from fitting the suscep-tibility vs. temperature data). One can clearly see (Fig. 3) that thetheoretical curves are in good agreement with the experimentalones.
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Fig. 4. Quantified EW as a function of magnetic field and temperature for Cu(tz)2Cl2. The magnetic field values are in Tesla and the temperature is in Kelvin.
Until now the quantification of entanglement has been donefrom susceptibility vs. temperature data. In this part of the Letterwe will use the experimental magnetization isotherms at differenttemperatures to estimate the amount of entanglement. The mag-netization isotherms as function of field enable us to capture thevariation of susceptibility and hence entanglement with tempera-ture and magnetic field. Magnetic susceptibility (χ ) is defined asthe field derivative of magnetization M (χ = ∂M
∂ B ), where B is themagnetic field. We have derived temperature dependent suscepti-bility at different fields using the above formula. These suscepti-bility datasets have been used to estimate EW. The experimentallyquantified EW as a function of magnetic field and temperature isshown in the 3D plot given in Fig. 4. It can be seen that entan-glement is decreasing with increasing temperature and vanishesat 12 K, compatible with the analysis done on the experimen-tal susceptibility vs. temperature data. From this plot one can seethat the EW at low temperature does not change significantly withthe change in magnetic field. This is owing to the strong couplingstrength between the spins in the spin chain. On application ofstrong magnetic field, the statistical weight of ferromagnetic statesincreases in the ensemble, thereby reducing the entanglement inthe system. This is tantamount to formation of triplet states at theexpense of singlet states. However, there are two competing en-ergy scales, the exchange coupling ‘ J ’ and the external magneticfield ‘B ’. To cause significant pair breaking, one needs to apply asignificantly higher magnetic field than was possible in our case.Thus we could not affect any substantial change in entanglementupon application of magnetic field.
4. Conclusion
With the advent of quantum information processing there isa renewed interest in spin chains especially where the groundstate is antiferromagnetic. This is because such a ground state isentangled for a spin 1/2 system. This entanglement can be wellcharacterized both theoretically and experimentally. We consideredone such spin 1/2 chain (Cu(tz)2Cl2) and characterized the entan-glement content from magnetic susceptibility data using an entan-glement witness. We also studied its variation both as a functionof temperature and magnetic field. The application of field as wellas increase in temperature cause a mixing in the states and thusthe ground state no longer remains an entangled pure state. Thusit is seen that the entanglement reduces upon increasing the tem-
perature. However, the magnetic field applied is not large enoughto cause significant amount of spin flip or pair breaking to reducethe entanglement at the lowest temperature and this is reflectedin our analysis. The entanglement obtained using the witness op-erator is the average entanglement between any two spins [13]. Inaddition to this we have also fitted our magnetic susceptibility andmagnetization data to the “Bonner–Fisher” model [21] to ascertainthat this compound is an antiferromagnetic spin 1/2 chain system.Our fits were excellent indicating that this is indeed an archetypalantiferromagnetic spin 1/2 system and a very suitable candidatefor the study of entanglement. This characterization of entangle-ment in spin chains can be useful in identifying suitable systemsfor quantum communication and also enable them to be useful asquantum networks connecting two quantum gates [14,15,25].
Acknowledgements
The authors would like to thank the Ministry of Human Re-source Development (MHRD), Government of India, for funding.The authors also would like thank to Dr. Swadhin Mandal for al-lowing us to use his lab facilities for the synthesis of the system.
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