Magnetization and magnetic phase diagrams of a spin-1/2 ferrimagnetic diamond chain at low temperature*

Tai-Min Cheng(成泰民), Mei-Lin Li(李美霖), Zhi-Rui Cheng(成智睿), Guo-Liang Yu(禹國梁),Shu-Sheng Sun(孫樹生), Chong-Yuan Ge(葛崇員), and Xin-Xin Zhang(張新欣)

1Department of Physics,College of Sciences,Shenyang University of Chemical Technology,Shenyang 110142,China

2School of Materials Science and Engineering,Shenyang University of Chemical Technology,Shenyang 110142,China

3School of Software Engineering,Shenyang University of Technology,Shenyang 110870,China

Keywords: invariant eigen-operator method,Jordan–Wigner transformations,critical magnetic field intensity,magnetic phase diagrams

1. Introduction

Low-dimensional frustration quantum magnetic systems have attracted much attention since they exhibit strange quantum phenomena, which are closely related to the lowtemperature quantum phase transition of low-dimensional condensed matter.[1–8]Especially in the diamond-chain compound Cu3(CO3)2(OH)2single crystal,[2]the 1/3 magnetization plateau and double peaks of magnetic susceptibility and specific heat were observed experimentally. This triggered intensive research interests in such low-dimensional magnetic chain materials.[3–7]Rule et al.[5]used inelastic neutron scattering experiments to investigate the exchange integrals of the diamond chain compound azurite Cu3(CO3)2(OH)2and again confirmed the existence of the 1/3 magnetization plateau from the experiment. Jeschke et al.[6]theoretically investigated the magnetic properties and the exchange integrals of the spin-1/2 diamond chain with density functional theory and explained the formation mechanism of the frustrated material with first-principles. In the calculation of properties in low-dimensional magnetic quantum systems, the densitymatrix renormalization group[9](DMRG)method exhibits its advantages. For example, the DMRG method is employed in the phase diagram[3,4]study of the one-dimensional magnetic quantum chain system.

In addition, there are also other theoretical methods to study the magnetic and thermodynamic properties of onedimensional magnetic chain,e.g.,the transfer-matrix densitymatrix renormalization group method[7,8](TMRG), the invariant eigen-operator (IEO) method[10–12]combined with Jordan–Wigner (J–W) transformations,[13,14]and the spinwave theory[15]method from a fully polarized vacuum combined with the quantum Monte Carlo (QMC) simulation.[15]The magnetic-field vs temperature phase diagram of alternate ferrimagnetic chains reported in the paper of Silva and Montenegro-Filho[15]has caught our attention. In their work,the relationship between the critical magnetic field and the double peak of magnetic susceptibility,as well as the correlation between the magnetization curve and the magnetic phase diagram,have attracted our great interest.

We have specifically investigated the properties of a spin-1/2 ferrimagnetic diamond XY chain[10]with the IEO method,two peaks in magnetic susceptibility and specific heat as well as two valleys in entropy appear at the critical magnetic field,which are similar to Silva’s work,[15]and the two peaks are also very similar in height.This triggered our motivation to investigate the magnetic phase diagram of the one-dimensional magnetic chain system with the IEO method.

The elementary excitation of the spin-1/2 ferrimagnetic chain system is also a fermion-type collective excitation that satisfies the Pauli exclusion principle. This fermion-type collective excitation is closely related to the Luttinger liquid(LL)phase in one-dimensional quantum wires. The quantum critical point separates a gapped insulating phase and a gapless conducting phase. The quantum critical state can be experimentally accessed through an applied magnetic field in magnetic insulators.[16,17]In these systems,the gapped phases are associated with magnetization plateaus in the magnetization curves. Oshikawa et al.[18]clarified that this magnetization plateau should be understood as a topological effect.

In this paper,the alternating ferrimagnetic diamond chain of the one-dimensional spin-1/2 Heisenberg with long-range interaction in the external magnetic field was investigated using the IEO method[10–12]combined with J–W transformations. Therefore,we derived the elementary-excitation energy spectra and partition function of the system under the external magnetic field. We only conducted a detailed study of the magnetization and the magnetic phase diagram of the system at absolute zero and finite temperature. Given the partition function of the system under the external magnetic field, it is easy to obtain the expressions of additional thermodynamic and magnetic physical quantities of the system (such as specific heat,internal energy,free energy,entropy,magnetic susceptibility).

2. The partition function and elementary excitations of the system

As shown in Fig. 1, dimer and monomer are alternately arranged to form a ferrimagnetic diamond chain with 1/2 spin.Here Sl,d1and Sl,d2represent the dimer’s spin quantum, and Sl,mrepresents the monomer’s spin quantum. Suppose that such a system has an N (l =1,2,...,N) cyclic structure. We use the Heisenberg interaction theory model to describe, and the Hamiltonian of the system can be expressed as follows:

Fig.1. The alternating ferrimagnetic diamond chain of the one-dimensional spin-1/2 Heisenberg model is composed of dimers and monomers.

The lattice Fourier transforms for operators al, bland clare as follows:

From Eqs. (4) and (17), the partition function of the system can be easily obtained as follows:

in which h=guBH.

3. Analysis and discussion

From the temperature-dependent thermal disorder energy(kBT), the Zeeman energy (guBH) that related to the external magnetic field, the ferromagnetic exchange energy (J2)that makes the electron spins in the dimer molecule parallel to each other, the antiferromagnetic exchange energy (J1and J3) that makes the electron spins between dimers and monomer molecules antiparallel arranged and the ferromagnetic exchange energy (Jm) which makes the electron spins of monomers and monomer molecules aligned in parallel,etc.Using the competition between the above energies, the magnetization behavior of the alternating ferrimagnetic diamond chain with a spin of 1/2 under an external magnetic field was analyzed.

To this end, when we change the exchange integral (J1,J2, J3, Jm), we study the change of the system’s magnetization with the external magnetic field at absolute zero and finite temperature. Thus,three critical magnetic field intensities HCB,HCEand HCSare obtained. Here HCBrepresents the critical magnetic field intensity,at which the system begins to exhibit a 1/3 magnetization plateau, HCErepresents the critical magnetic field intensity where the 1/3 magnetization plateau disappears, and HCSrepresents the critical magnetic field intensity where the system begins to exhibit the fully polarized(FP)plateau.

According to Eqs. (13) and (14), the elementaryexcitation spectra of the system satisfies the relationship of ?ω1,k＞?ω3,k＞?ω2,kin the entire Brillouin zone (where ?ω1,k＞0,but ?ω3,kand ?ω2,kcan be positive or negative).

When H ≥HCS, the energies of three elementary excitations are greater than zero in the Brillouin zone, and they drive the system from low energy state to high energy state,leading to an energy gap(UP gap)and the fully polarized(FP)plateau phase. FP phase appears when H is higher than HCS.In this area,the magnetic moment of dimer molecules and the single matrix molecule is fully parallel to the external magnetic field,which is similar to that of the ferromagnetic phase.The whole system is fully polarized. When HCS＞H ＞HCE,the energy of ?ω2,kcan be positive or negative, resulting in a gapless state of the system and the Luttinger liquid(LL)phase.LL phase emerges when the magnetic field intensity H is in the range of HCE＜H ＜HCS. In this area,the magnetic moment of the monomer’s molecule will rotate along the direction of the external magnetic field when the elementary excitation energy increases, which will induce the magnetic moment of dimer’s molecules to deviate from the external magnetic field randomly. The projection of the total magnetic moment along the direction of the external magnetic field will increase with the external magnetic field.When HCE≥H ≥HCB,the energy of ?ω2,kappears to be negative. It transforms the system from a high energy state to a low energy state,leading to an energy gap(DN gap)and the ferrimagnetic(FRI,1/3 magnetization)plateau phase. FRI plateau phase appeared when the magnetic field intensity H is in the range of HCE≥H ≥HCB. In this area,the magnetic moment of the dimer’s molecules is parallel to the external magnetic field, whereas the magnetic moment of the monomer’s molecule is along the opposite direction.The total magnetic moment of the ferrimagnetic system is the maximum and remains unchanged with the increase of the external magnetic field. When HCB＞H,the energy of ?ω3,kcan be positive or negative,inducing a gapless state of the system and the ferrimagnetic(FRI)phase. FRI phase is a disordered ferrimagnetic-like phase in which the magnetic moments of the dimer’s molecule are arranged approximately antiparallel to the magnetic moments of the monomer’s molecule,but the size of the magnetic moment is different. Moreover, in this phase, the projection of the total magnetic moment along the direction of the external magnetic field will increase with the external magnetic field.

Combined with Eq.(20)and Fig.2,it can be seen that at absolute zero temperature,when the energy of the elementaryexcitation spectrum is greater than zero,the Fermi–Dirac statistical distribution 〈nσ〉=1/(1+e?ωσ,k/(kBT)), (σ=1,2,3)is zero. Due to ?ω1,k＞0,so〈n1〉=0. Moreover,the energy of the elementary-excitation spectra will increase with the increase of the external magnetic field intensity (see Fig. 2).

Fig.2. The dispersion curves of elementary-excitation spectrum at 0 K in different external magnetic fields(guBH/J=0.0,0.15,0.34,0.42,0.50,0.75,1.02,1.17)(other parameters are set as J1/J=0.47,J2/J=?1,J3/J=0.21,Jm/J=?0.1).

The elementary excitations (?ω1,k, ?ω2,kand ?ω3,k) satisfy the Fermi–Dirac statistical distribution. The analysis of elementary-excitation spectra gives the relationship with the critical magnetic field as follows.Here HCB,HCE,and HCSare the external magnetic field intensities as the minimum value of ?ω3,k, the maximum value of ?ω2,k, and the minimum value of ?ω2,kare equal to zero,respectively. This can be expressed by the following equations:

We use Eq.(21)to draw the magnetic phase diagrams of magnetic field-different exchange integrals at absolute zero.The relationships between the system’s magnetization curve at finite temperature and the critical external magnetic field intensities are obtained according to Eq.(20),and the magnetic field-temperature phase diagram is drawn.

In the magnetic phase diagrams, the FP plateau phase is represented in olive green, the LL phase is in blue, the FRI plateau phase(i.e.,1/3 magnetization plateau phase)is in magenta,and the FRI phase is in cyan(see Figs.4,6,9 and 12).

Fig.3.The magnetization change with the external magnetic field at different temperatures(other parameters are set as J1/J=0.47,J2/J=?1,J3/J=0.21,Jm/J=?0.1).

Fig. 4. The critical magnetic field intensities (HCB, HCE and HCS) as a function of temperature (other parameters are set as J1/J = 0.47,J2/J=1.0,J3/J=0.21,Jm/J=?0.1).

Fig. 5. The magnetization changes with the external magnetic field at T =0 K,under different ferromagnetic exchange integrals J2/J (?0.5,?0.75, ?1.0, ?1.25, ?1.5, ?1.75, ?2.0) (other parameters are set as J1/J=0.47,J3/J=0.21,Jm/J=?0.14).

It can be seen from Figs. 3 and 4 that in the spin-1/2 alternating ferrimagnetic diamond chain, when the reduced exchange integrals are J1/J=0.47, J2/J=?1, J3/J=0.21,Jm/J = ?0.1, the spin magnetic order of the system is destroyed by the thermal disorder energy with the increase of temperature.Therefore,the symmetry of the system increases,and the order decreases. Its manifestation is in the low temperature range of kBT/J ＜0.008,the width of the 1/3 magnetization plateau of the system gradually decreases with the increase of temperature(see Figs.3 and 4). The width of the 1/3 magnetization plateau is the largest at 0 K.As the temperature increases, the critical magnetic field intensity HCBgradually increases(i.e.,moves to the right),while the critical magnetic field intensity HCEdecreases(i.e.,moves to the left),resulting in the width of 1/3 magnetization plateau gradually decreasing(see Fig.3). Until kBT/J=0.008, the 1/3 magnetization plateau completely disappears. But the region of the spin-flop states(i.e.,LL phase)gradually increases with increasing temperature. When kBT/J ＞0.008,the critical magnetic field intensity HCBand HCEof the system are completely coincident and are independent of temperature.The above analysis shows that the appearance of the 1/3 magnetization plateau of the system is closely related to the quantum effect of the system.Because the quantum effect of the system is more pronounced at extremely low temperatures, but this quantum effect is not observed at room temperature. This is because the quantum effects are completely suppressed by the thermal disorder energy at room temperature. Therefore, the appearance of the 1/3 magnetization plateau in the low temperature region is the macroscopic manifestation of the quantum effect. The critical temperature at which the 1/3 magnetization plateau disappears is kBT/J =0.008, this temperature corresponds to the threephase critical point of the FRI phase, the FRI plateau phase,and the LL phase(see Fig.4).

Fig.6. The critical magnetic field intensities(HCB,HCE and HCS)as a function of ?J2/J at T =0 K(other parameters are set as J1/J=0.47,J3/J=0.21,Jm/J=?0.14).

Fig.7.The magnetization changes with the external magnetic field at finite temperature kBT/J=0.01,under different ferromagnetic exchange integrals J2/J (?0.5, ?0.75, ?1.0, ?1.25, ?1.5, ?1.75, ?2.0)(other parameters are set as J1/J=0.47,J3/J=0.21,Jm/J=?0.14).

As can be seen from Figs. 5–7, when the reduced exchange integrals (J1/J =0.47, J3/J =0.21, Jm/J =?0.14),with the increase of reduced ferromagnetic exchange integral?J2/J (0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0), the critical magnetic field intensity HCEdecreases (i.e., moves to the left) at T =0 K and finite temperature.However,the critical magnetic field intensities HCBand HCSdo not change,which means that HCBand HCShave nothing to do with the ferromagnetic exchange energy(J2)that makes the electron spins of the dimer molecules parallel. Therefore, as the reduced ferromagnetic exchange integral ?J2/J increases,the region of the spin-flop states expands,and the width of the 1/3 magnetization plateau gradually decreases, but the decreasing rate becomes slower and slower and consequently, leading to the width of the 1/3 magnetization plateau tend to be constant. The thermal disorder energy kBT destroys the magnetic order of the system(see Fig. 7), with the increase of the reduced ferromagnetic exchange integral ?J2/J, which accelerates the expansion of the region of spin-flop states and decreases the width of the 1/3 magnetization plateau.There is no three-phase critical point in the magnetic phase diagram, so the change of ferromagnetic exchange integral ?J2/J does not cause the disappearance of 1/3 magnetization plateau(see Fig.6).

In Fig. 8, when the reduced exchange integrals are J2/J=?1,J3/J=0.21,Jm/J=?0.14,and the external magnetic field guBH/J=0,and the reduced antiferromagnetic exchange integral J1/J=0, we can be obtained from Eqs. (13)and(14)that there are ?ω1,k＞0,?ω3,k＞0,and ?ω2k＜0 in the entire Brillouin zone. And elementary excitations ?ω1,k,?ω2,kand ?ω3,ksatisfy the Fermi–Dirac statistical distribution.Therefore,from Eq.(20),it can be seen that when the external magnetic field guBH/J=0, and J1/J=0, the total magnetization M/(guBN)=0.5 at T =0 K(see Fig.8(a)). According to the principle of symmetry, the same result at 0 K should be obtained when J2/J = ?1, J1/J = 0.21, Jm/J = ?0.14,guBH/J =0 and J3/J =0. However, as can be seen from Fig. 10(a), when the reduced exchange integrals are J2/J =?1.0,J1/J=0.47,Jm/J=?0.14,the external magnetic field guBH/J=0,and the reduced antiferromagnetic exchange integral J3/J =0, we can obtain from Eqs. (13) and (14) that there are ?ω1,k＞0,?ω3,k＜0 and ?ω2,k＜0 in the entire Brillouin zone. So, from Eq. (20), we can know that when the external magnetic field guBH/J =0, and J3/J =0, the total magnetization M/(guBN)=?0.5 at T =0 K.Similarly,when J2/J =?1.0, J3/J =0.47, Jm/J =?0.14, guBH/J =0 and J3/J=0,the same result should be obtained.

Fig. 8. The magnetization changes with the external magnetic field at(a)T =0 K and(b)finite temperature kBT/J=0.02, under different antiferromagnetic exchange integrals J1/J (0.0, 0.1, 0.2, 0.33, 0.4, 0.5, 0.6)(other parameters are set as J2/J=?1,J3/J=0.21,Jm/J=?0.14).

It can be seen from Figs. 8 and 10 that the critical magnetic field intensities HCB,HCEand HCSincrease(all move to the right) with the increase of the reduced antiferromagnetic exchange integrals (J1/J, J3/J), and the region of spin-flop states widens.However,if only J1/J is changed(see Figs.8(a)and 9), the 1/3 magnetization plateau disappears when the J1/J=0.33 at 0 K.As can be seen from Figs.7(b)and 9(b),thermal disorder energy kBT accelerates the decrease and the increase of the width of the 1/3 magnetization plateau and the region of spin-flop states,respectively.

Fig. 9. The critical magnetic field intensities (HCB, HCE and HCS) as a function of J1 at T =0 K (other parameters are set as J2/J =?1.0,J3/J=0.21,Jm/J=?0.14).

As can be seen from Fig.9,in a spin-1/2 alternating ferrimagnetic diamond chain at T = 0 K, when J2/J = ?1.0,J3/J = 0.21, Jm/J = ?0.14, and the antiferromagnetic exchange integral J1/J ＜0.061, as the antiferromagnetic exchange integral J1increases, the width of the 1/3 magnetization plateau of the system increases more and more. And with the increase of J1,the saturation critical magnetic field intensity HCSand HCEincreases linearly, and the increase in the amplitude and rate of HCSis greater than that of HCE, so the region of spin-flop states has expanded. This is because the critical magnetic field intensity HCBdoes not change with J1in this range (J1/J ＜0.061). When 0.061 ＜J1/J ＜0.33, as the antiferromagnetic exchange integral J1increases,the critical magnetic field intensity HCBbegins to increase linearly,and the rate of increase is greater than that of HCE. As a result,the 1/3 magnetization plateau gradually decreases until it completely disappears near J1/J =0.33. When J1/J ＞0.33,the 1/3 magnetization plateau and the region of spin-flop states gradually increase with the increase of J1. Here,the 1/3 magnetization plateau disappears at the three-phase critical point of the FRI phase,the FRI plateau phase,and the LL phase.The critical antiferromagnetic exchange integral corresponding to this three-phase critical point is J1/J=0.33(see Fig.9).

Fig.10. The magnetization changes with the external magnetic field at(a) T =0 K and (b) finite temperature kBT/J =0.02, under different anti-ferromagnetic exchange integrals J3/J (0.0, 0.2, 0.4, 0.8) (other parameters are set as J2/J=1,J1/J=0.47,Jm/J=0.14).

It can be seen from Figs.11 and 12 that when the reduced exchange integral J1/J=0.47,J2/J=?1,J3/J=0.21 is unchanged, at T =0 K (see Figs. 11(a), 11(b) and 12), when?Jm/J ＜0.184, the critical magnetic field intensity HCEdecreases (i.e., moves to the left) with the ferromagnetic exchange integral ?Jm/J increase. However, the critical magnetic field intensities HCBand HCSdo not change, resulting in the width of 1/3 magnetization plateau gradually decreasing and the region of spin-flop states gradually expanding.When ?Jm/J =0.184, the 1/3 magnetization plateau disappears completely. When 0.336 ＞?Jm/J ＞0.184, the critical magnetic field intensity HCBdecreases (i.e., moves to the left),but the critical magnetic field intensities HCEand HCSdo not change,resulting in the width of 1/3 magnetization plateau gradually increasing,and the region of spin-flop states are constant and independent of Jm. When ?Jm/J ＞0.366,the three critical magnetic field intensities HCB, HCEand HCSare constant and independent of the ferromagnetic exchange integral Jm. Therefore, when ?Jm/J ＞0.366, the width of the 1/3 magnetization plateau and the width of the region of spin-flop states are constant and independent of Jm. Here,the 1/3 magnetization plateau disappears at the three-phase critical point of the FRI phase, the FRI plateau phase, and the LL phase.The critical ferromagnetic exchange integral corresponding to this three-phase critical point is ?Jm/J=0.184(see Fig.12).

When ?Jm/J = 0.184, the 1/3 magnetization plateau completely disappears, and under the effect of thermal disorder energy kBT outside the FP plateau region, this magnetization curve is separated from the magnetization curve in other cases (Jm/J =0.0, ?0.05, ?0.1, ?0.15, ?0.25, ?0.3,?0.35)at the finite temperature kBT/J=0.01 except for the FP plateau region (see Figs. 11(c) and 11(d)). This shows that there is a competition between magnetic order, which is caused by an external magnetic field in the system, and thermal disorder at finite temperatures.

Fig.11. The magnetization changes with the external magnetic field at(a), (b)T =0 K and(c), (d)finite temperature kBT/J=0.01, under different ferromagnetic exchange integrals Jm/J (0.0,?0.05,?0.1,?0.15,?0.184,?0.25,?0.3,?0.35)(other parameters are set as J1/J=0.47,J2/J=?1.0,J3/J=0.21).

Fig.12. The critical magnetic field intensities(HCB,HCE and HCS)as a function of Jm/J at T =0 K(other parameters are set as J1/J=0.47,J2/J=?1,J3/J=0.21).

4. Conclusion

We have obtained the following conclusions by studying the magnetization and magnetic phase diagram of the spin-1/2 alternating ferrimagnetic diamond chain in external magnetic fields at finite temperatures.

(2) The 1/3 magnetization plateau will inevitably disappear as long as the three-phase critical point exists in the magnetic phase diagram,which is intersected by the FRI phase,the FRI plateau phase,and the LL phase. However,the 1/3 magnetization plateau will not disappear without this three-phase critical point.

(3)The UP and the DN energy gap induce the appearance of the FP and the FRI plateau phase,respectively. Also,?ω2,kand ?ω3,kcan be positive or negative in the Brillouin zone,resulting in a gapless state in the system, then inducing the appearance of the LL and the FRI phase,respectively.

(4) The width of the 1/3 magnetization plateau is the largest at 0 K. As the temperature increases, HCBgradually increases, while HCEdecreases, resulting in the width of 1/3 magnetization plateau gradually decreasing. Therefore,the appearance of the 1/3 magnetization plateau in the lowtemperature region is the macroscopic expression of the quantum effect.

(5)The change of the ferromagnetic exchange energy ?J2does not change HCBand HCS. However,HCEdecreases with the increase of ?J2. As a result,the width of the 1/3 magnetization plateau decreases and the region of spin-flop states gets enlarged.

(6) There are two critical values (?JmC1and ?JmC2)for the ferromagnetic exchange integral ?Jm. When ?Jm＜?JmC1, as ?Jmincreases, HCEdecreases, but HCBand HCSremain unchanged, resulting in the width of the 1/3 magnetization plateau decreasing and the region of spin-flop states enlarged. When ?Jm=?JmC1,the 1/3 magnetization plateau disappears completely.When ?JmC2＞?Jm＞?JmC1,as ?Jmincreases,HCBdecreases,but HCEand HCSremain unchanged,resulting in the width of the 1/3 magnetization plateau gradually increases, while the region of spin-flop states is unchanged.When ?Jm＞?JmC2,the three critical magnetic field intensities HCB,HCEand HCSare not related to ?Jm.

(7) HCB, HCE, and HCSincrease with the antiferromagnetic exchange integral(J1or J3). While the change of J1or J3can tune the 1/3 magnetization plateau,so that the width of the 1/3 magnetization plateau of the system changes from large to small until it disappears,and then increases. The above results indicated that the appearance of the 1/3 magnetization plateau is closely related to the magnetic ordered structure of the system.