A coupled spin-electron diamond chain with different Landé g-factors of localized Ising spins and mobile electrons
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] S e p A coupled spin-electron diamond chain with different Land´e g-factors of localizedIsing spins and mobile electrons
Jordana Torrico , Maria Socorro Seixas Pereira , Jozef Streˇcka , Marcelo Leite Lyra Instituto de F´ısica, Universidade Federal de Alagoas, 57072-970 Macei´o, Alagoas, Brazil and Department od Theoretical Physics and Astrophysics, Faculty of Science,P. J. ˇSaf´arik University, Park Angelinum 9, 040 01 Koˇsice, Slovakia
A coupled spin-electron diamond chain with localized Ising spins placed on its nodal sites andmobile electrons delocalized over interstitial sites is explored in a magnetic field taking into accountthe difference between Land´e g-factors of the localized spins and mobile electrons. The ground-state phase diagram is constituted by two classical ferrimagnetic phases, the quantum unsaturatedparamagnetic phase and the saturated paramagnetic phase. Both classical ferrimagnetic phases aswell as the unsaturated paramagnetic phase are reflected in a low-temperature magnetization curveas intermediate magnetization plateaus. The unsaturated paramagnetic phase is quantum in itscharacter as evidenced by the fermionic concurrence calculated for a pair of the mobile electronshopping in between the interstitial sites. It is shown that the magnetic field can under certainconditions induce a quantum entanglement above the disentangled ground state.
PACS numbers: 05.50.+q, 75.10.Pq, 75.30.Kz, 75.40.Cx
I. INTRODUCTION
Frustrated spin systems exhibit a variety of exoticquantum ground states, which may provide an inter-esting alternative for a quantum information process-ing [1]. The geometric spin frustration is most com-monly introduced through competing antiferromagneticinteractions between the localized spins situated on non-bipartite lattices. Another remarkable alternative repre-sents a kinetically-driven spin frustration of the localizedspins, which is invoked by a quantum-mechanical hop-ping process of the mobile electrons. The latter type ofspin frustration has been found for instance in the cou-pled spin-electron diamond chain [2–5]. Last but notleast, recent studies motivated by a magnetic behavior ofthe copper-iridium oxides have revealed another peculiarmechanism of the spin frustration, which originates froma non-uniformity of the Land´e g-factor [6].In the present work, we will explore a combined ef-fect of the kinetically-driven spin frustration and thespin frustration stemming from the non-uniformity of theLand´e g-factors by generalizing the exact solution for thecoupled spin-electron diamond chain [2–5]. Following theprocedure elaborated in our previous work [5] we willcompute the fermionic concurrence between the pair ofmobile electrons in order to demonstrate how the mag-netic field may induce a quantum entanglement abovethe disentangled zero-field ground state.
II. MODEL AND METHOD
Let us consider a coupled spin-electron diamond chain,which is composed of the localized Ising spins situated onits nodal lattice sites and two mobile electrons hoppingon the pairs of interstitial sites (see Fig. 1 of Ref. [5]for illustration). The total Hamiltonian of the model under investigation can be written as a sum of the cellHamiltonians H = P i H i , whereas each cell Hamiltonian H i involves all the interaction terms belonging to the i -thdiamond unit: H i = − t X γ = ↑ , ↓ (cid:16) c † i ,γ c i ,γ + h . c . (cid:17) − g h X j =1 ( n ij, ↑ − n ij, ↓ )+ J ( σ i + σ i +1 ) X j =1 ( n ij, ↑ − n ij, ↓ ) − g h σ i + σ i +1 ) . (1)Here, c † iα,γ and c iα,γ are fermionic creation and annihi-lation operators for an electron with the spin γ = ↑ , ↓ hopping on the pairs of interstitial sites α = 1 , n iα,γ = c † iα,γ c iα,γ is respective number operator. The parameter t is the hopping term associated with the kinetic energyof the mobile electrons, the coupling constant J standsfor the Ising interaction between the nearest-neighborlocalized spins and mobile electrons, h is the externalmagnetic field accounting for the difference between theLand´e g-factors g and g of the mobile electrons and lo-calized Ising spins, respectively (Bohr magneton µ B wasabsorbed into a definition of the field term h ).An exact solution for the coupled spin-electron dia-mond chain can be straightforwardly obtained by adapt-ing the procedure reported in our previous work [5] by amere replacement of the uniform magnetic field h throughtwo different local magnetic fields g h and g h . The read-ers interested in details of the calculation procedure aretherefore referred to Ref. [5]. III. RESULTS AND DISCUSSION
Let us proceed to a discussion of the most interestingresults for the coupled spin-electron diamond chain with
UPASPAFRI h / J (a)(c) UPASPAFRI h / J t/J (d) FRI UPASPA t/J (b)
UPASPA
FRI Figure 1: The ground-states phase diagram in the t/J - h/J plane for the fixed value of the Land´e g-factor of the mobileelectrons g = 2 and a few different values of the Land´e g-factor of the localized Ising spins: (a) g = 1 .
8; (b) g = 2;(c) g = 3; (d) g > the antiferromagnetic exchange interaction J >
0, whichexhibits the most outstanding magnetic features due to aspin-frustration effect. For simplicity, our further atten-tion will be restricted to the most common special casewith the fixed value of the Land´e g-factor of the mobileelectrons g = 2, whereas an influence of the Land´e g-factor of the localized Ising spins on the overall magneticbehavior will be subject of our investigations. A. Ground-state phase diagram
The ground-state phase diagram (Fig. 1) involves twoferrimagnetic phases (FRI and FRI ), the unsaturatedparamagnetic phase (UPA) and the saturated paramag-netic phase (SPA) given by the eigenvectors: | FRI i = N Y i =1 |↓i σ i ⊗ |↑ , ↑i i (2) | FRI i = N Y i =1 |↑i σ i ⊗ |↓ , ↓i i (3) | UPA i = N Y i =1 |↑i σ i ⊗
12 [ |↑ , ↓i i + |↓ , ↑i i − |↑↓ , i i − | , ↑↓i i ] (4) | SPA i = N Y i =1 |↑i σ i ⊗ |↑ , ↑i i . (5) In above, the first ket vector determines the spin state oflocalized Ising spins and the second one the spin state ofthe mobile electrons. All magnetic moments of the local-ized Ising spins and mobile electrons are fully aligned intothe magnetic field within the SPA ground state. Owingto the antiferromagnetic interaction J >
0, the localizedIsing spins tend in opposite to the magnetic field withinthe FRI ground state and the mobile electrons tend inopposite to the magnetic field within the FRI groundstate. However, the most interesting spin arrangementcan be found within the UPA ground state, where thehopping process of two mobile electrons with oppositespins leads to a kinetically-driven spin frustration of thelocalized Ising spins. As a result, the localized Ising spinsare polarized by arbitrary but non-zero magnetic field,while the mobile electrons underlie a quantum entangle-ment of two antiferromagnetic and two ionic states. Notefurthermore that the FRI (FRI ) phase appears in theground-state phase diagram if g < g > g = 4. B. Magnetization curves
To get a deeper insight, the total magnetization is plot-ted in Fig. 2 against the magnetic field together withthe sublattice magnetization of the Ising spins and themobile electrons. The magnetization dependences shownin the first column give evidence for the following se-quence of the phase transitions FRI -UPA-SPA drivenby the rising magnetic field. Contrary to this, the totaland sublattice magnetizations plotted in the second col-umn give proof for another sequence of the field-inducedphase transitions FRI -UPA-SPA. The displayed magne-tization curves thus independently verify correctness ofthe established ground-state phase diagram. C. Fermionic concurrence
The fermionic concurrence C , which may serve as ameasure of bipartite entanglement between two mobileelectrons from the same couple of interstitial sites, canbe calculated at zero as well as nonzero temperatures ac-cording to the procedure described in Ref. [5]. The clas-sical character of the FRI , FRI and SPA ground statesis consistent with zero concurrence C = 0, while its max-imum value C = 1 reveals a full quantum entanglementwithin the UPA ground state. Typical thermal depen-dences of the concurrence are illustrated in Fig. 3. Ingeneral, the concurrence monotonically decreases from itsmaximum value C = 1 with increasing temperature whenthe interaction parameters drive the investigated systemtowards the UPA ground state. In addition, the concur-rence exhibits a more striking non-monotonous thermaldependence if the magnetic field drives the investigatedsystem sufficiently close to a phase boundary betweenthe UPA ground state and one of three classical (FRI , (e) M / M S h/J (f) k B T/J=0.0 k B T/J=0.2 k B T/J=0.5 k B T/J=1.0 h/J (c) M e [ g ] (d) M [ g ] (b) (a) Figure 2: (Color online) The sublattice magnetization of theIsing spins M σ , the sublattice magnetization of the mobileelectrons M e and the total magnetization as a function ofmagnetic field for t/J = 1, g = 2 and a few different valuesof temperature. The first column corresponds to g = 3 andthe second column to g > FRI and SPA) ground states. Under this condition, theconcurrence shows a peculiar reentrant behavior due toa thermally-induced activation of the UPA spin arrange-ment, which represents low-lying excited state above oneof three classical (FRI , FRI and SPA) ground states. IV. CONCLUSION
In the present work, we have generalized an exact so-lution for a coupled spin-electron diamond chain by ac-counting for a difference between the Land´e g-factorsof the localized Ising spins and mobile electrons. Theground-state phase diagram, magnetization process and fermionic concurrence have been investigated in partic-ular. It has been verified that the ground-state phasediagram involves two classical ferrimagnetic phases FRI and FRI , the quantum unsaturated paramagnetic phaseUPA as well as the saturated paramagnetic phase SPA.The ground states FRI , FRI and UPA manifest them-selves in a low-temperature magnetization curve as inter-mediate magnetization plateaus. The quantum characterof the UPA ground state has been evidenced through thefermionic concurrence, which displays monotonous de- h/J=0.95 h/J=1.00 h/J=1.25 h/J=1.60 C (a) C k B T/J (c) h/J=0.5 h/J=1.0 h/J=2.0 h/J=2.5 (b) h/J=0.4 h/J=0.5 h/J=1.0 h/J=1.6 (d) h/J=0.5 h/J=1.0 h/J=2.0 h/J=2.5 k B T/J
Figure 3: (Color online) Thermal variations of the concur-rence at a few different magnetic fields when g = 2 is fixedand: (a) t/J = 1, g = 3; (b) t/J = 1, g >
4; (c) t/J = 2, g = 3; (d) t/J = 2, g > cline upon rising temperature. It has been also demon-strated that the nonzero fermionic concurrence can beinduced above the classical ground states once the mag-netic field drives the investigated system sufficiently closeto a phase boundary with the UPA ground state. Acknowledgments
This work was supported by Brazilian agencies FA-PEAL, CNPq, CAPES and by Slovak Research and De-velopment Agency under contract No. APVV-0097-12. [1] M.A. Nielsen, I.L. Chuang,
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