International Scholarly Research Notices

International Scholarly Research Notices / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 980192 |

M. R. Soltani, J. Vahedi, M. R. Abolhassani, A. A. Masoudi, "Heisenberg Model with Added Dzyaloshinskii-Moria Interaction", International Scholarly Research Notices, vol. 2011, Article ID 980192, 6 pages, 2011.

Heisenberg Model with Added Dzyaloshinskii-Moria Interaction

Academic Editor: W. Selke
Received25 Oct 2011
Accepted30 Nov 2011
Published05 Jan 2012


We have considered the 1D spin-(1/2) Heisenberg model with added Dzyaloshinskii-Moriya interaction. The effect of a uniform magnetic field on the ground state phase diagram of the model is studied. We have mapped the model to an effective model which is known as the 1D XXZ in both uniform and staggered magnetic fields. By selecting a block of two or three spins, we have solved the Hamiltonian exactly. Our results show that the quantum phase transitions can be obtained from the block of pair or three spins.

1. Introduction

Study of the magnetic field effect on the ground state characteristics of chain model of antiferromagnetic (AF) spin-(1/2) has attracted much interest in recent years. The Hamiltonian of this model in a homogenous magnetic field ℎ is given by 𝐻=𝐽𝑁𝑗=1𝑆𝑥𝑗𝑆𝑥𝑗+1+𝑆𝑦𝑗𝑆𝑦𝑗+1+𝑆𝑧𝑗𝑆𝑧𝑗+1+â„Žğ‘î“ğ‘—=1𝑆𝑥𝑗,(1) where 𝐽>0 is the exchange coupling and ℎ denotes the magnetic field. The exact solution for ℎ=0 is obtained by Bethe ansatz [1]. The energy spectrum is gapless and the system is in the Luttinger liquid phase. In this phase, the spin correlation functions are in power form. When the system imposes by homogeneous magnetic field, its spectrum remains gapless up to the critical field â„Žğ‘=2𝐽. Here the Pokrovsky-Talapov phase transition takes place and the ground state of the system becomes saturated ferromagnetic [2, 3]. In many experimental data, the resulting data are different by theoretical predications [4–10]. These differences are due to DM interaction [11, 12], its Hamiltonian can be written as follows: 𝐻=𝐽𝑁𝑗=1𝑆𝑥𝑗𝑆𝑥𝑗+1+𝑆𝑦𝑗𝑆𝑦𝑗+1+𝑆𝑧𝑗𝑆𝑧𝑗+1+𝑁𝑗=1⃗𝑆𝐷⋅𝑗×𝑆𝑗+1+â„Žğ‘î“ğ‘—=1𝑆𝑥𝑗,(2) where ⃗𝐷 is the DM vector and chooses in the 𝑧 direction. [As DM interaction broken the fundamental SU(2) symmetry which is related to Heisenberg isotropic interaction]. It is known as the origin of many declination and creates many different qualitative effects. It specially creates an energy gap by the scale of gap≈(ğ·â„Ž)2/3 [13, 14]. DM interaction can act as a vector potential on the spin wave in the magnon spin Hall effect [15]. In ferromagnetic nanowires DM interaction has profound effect on the motion of domain walls [16]. It can also give rise to spin current and soliton in spin chains [17]. We emphasis that the studies of Heisenberg model commonly without DM interaction and the role of DM interaction on GS of Heisenberg model of AF by spin-(1/2) are considered less than the other models. As the exact solution of this model with DM interaction cannot be done, we need to do a great value of theoretical works.

2. Mapping to the XXZ Chain

In this section we try to map the Heisenberg chain in the presence of external magnetic field and DM interaction to the well-known XXZ chain, analytically. At first we have done a 𝜋/2 rotation around X, to convert 𝑧→𝑦,𝑦→𝑧:𝐻=𝐽𝑁𝑗=1𝑆𝑥𝑗𝑆𝑥𝑗+1+𝑆𝑦𝑗𝑆𝑦𝑗+1+𝑆𝑧𝑗𝑆𝑧𝑗+1+𝐷𝑁𝑗=1𝑆𝑥𝑗𝑆𝑦𝑗+1−𝑆𝑦𝑗𝑆𝑥𝑗+1+â„Žğ‘î“ğ‘—=1𝑆𝑥𝑗.(3)

Using 𝑆± operators [15], 𝑆±𝑗=𝑆𝑥𝑗±𝑆𝑦𝑗⟹𝑆𝑥𝑗=12𝑆+𝑗+𝑆−𝑗,𝑆𝑦𝑗=1𝑆2𝑖+𝑗−𝑆−𝑗.(4)

The interacting Hamiltonian terms can be calculated as follows:𝑆𝑥𝑗𝑆𝑦𝑗+1=1𝑆4𝑖+𝑗+𝑆−𝑗𝑆+𝑗+1−𝑆−𝑗+1=1𝑆4𝑖+𝑗𝑆+𝑗+1−𝑆+𝑗𝑆−𝑗+1+𝑆−𝑗𝑆+𝑗+1−𝑆−𝑗𝑆−𝑗+1,𝑆𝑦𝑗𝑆𝑥𝑗+1=1𝑆4𝑖+𝑗−𝑆−𝑗𝑆+𝑗+1+𝑆−𝑗+1=1𝑆4𝑖+𝑗𝑆+𝑗+1+𝑆+𝑗𝑆−𝑗+1−𝑆−𝑗𝑆+𝑗+1−𝑆−𝑗𝑆−𝑗+1,𝑆𝑥𝑗𝑆𝑥𝑗+1=14𝑆+𝑗+𝑆−𝑗𝑆+𝑗+1+𝑆−𝑗+1=14𝑆+𝑗𝑆+𝑗+1+𝑆+𝑗𝑆−𝑗+1+𝑆−𝑗𝑆+𝑗+1+𝑆−𝑗𝑆−𝑗+1,𝑆𝑦𝑗𝑆𝑦𝑗+1=1𝑆4𝑖+𝑗−𝑆−𝑗𝑆+𝑗+1−𝑆−𝑗+1=14−𝑆+𝑗𝑆+𝑗+1+𝑆+𝑗𝑆−𝑗+1+𝑆−𝑗𝑆+𝑗+1−𝑆−𝑗𝑆−𝑗+1,𝑆𝑥𝑗𝑆𝑥𝑗+1+𝑆𝑦𝑗𝑆𝑦𝑗+1=12𝑆+𝑗𝑆−𝑗+1+𝑆−𝑗𝑆+𝑗+1.(5) Substituting these terms, we simply obtain the following: 𝐻DM=𝐷𝑁𝑗=1𝑆𝑥𝑗𝑆𝑦𝑗+1−𝑆𝑦𝑗𝑆𝑥𝑗+1=𝐷2𝑖𝑁𝑗=1𝑆−𝑗𝑆+𝑗+1−𝑆+𝑗𝑆−𝑗+1,𝐻XX𝑥=𝐽2𝑁𝑗=1𝑆+𝑗𝑆−𝑗+1+𝑆−𝑗𝑆+𝑗+1+2𝑆𝑧𝑗𝑆𝑧𝑗+1.(6) Finally one can find the effective Hamiltonian:𝐻1=𝐽cos𝜑𝑁𝑗=1𝑆𝑥𝑗𝑆𝑥𝑗+1+𝑆𝑦𝑗𝑆𝑦𝑗+1+cos𝜑𝑆𝑧𝑗𝑆𝑧𝑗+1î‚âˆ’â„Žğ‘î“ğ‘—=1ğ‘†ğ‘¥ğ‘—âˆ’â„Žâ€²ğ‘î“ğ½=1(−1)𝑗𝑆𝑦𝑗,(7) in which âƒ—âƒ—â„Žâ„Žâ€²âˆğ·Ã— is known as staggered field and 𝐷=𝐽tan𝜑. This Hamiltonian is considered as XXZ chain Hamiltonian in homogenous and staggered field. It can be seen that AF can be created by DM interaction. It is shown theoretically that applying a staggered field on the AF chain of Heisenberg spin-(1/2), causes an energy gap in the spectrum of the system. In the absences of homogenous and staggered magnetic field (ℎ=ℎ′=0) the GS of the system is in the spin Luttinger liquid phase. In the presence of homogenous field, the exact solution is obtained by the Bethe ansatz methods. In the presence of staggered magnetic field, the exact solution is impossible. AF causes Neel ordered in 𝑦 direction (unrotated z direction). When ℎ′=0,ℎ≠0, the magnetization in the ℎ direction is an uniform and increasing function of ℎ.

In what follows, we will consider two- and three-particle systems because they can be exactly calculated, and this phenomena takes place in some alternative spin chain that gives us a qualitative view of real interacting model which cannot be solved.

3. Two Particles Spin-(1/2) Systems

In this section we consider a special model which composed of two particle systems. We choose AF interacting for spins. 𝑆𝐻=𝐽1⋅𝑆2𝑆+𝐷1×𝑆2î‚î€·ğ‘†âˆ’â„Žğ‘¥1+𝑆𝑥2.(8) To solve this, we choose the singlet and triplet states for Hilbert space. We know that its eigenstates relate to 𝑆𝑧1𝑆𝑧2 as follows: 𝑆𝑧1𝑆𝑧2||1↑↑⟩=4||↑↑⟩,𝑆𝑧1𝑆𝑧2||1↑↓⟩=−4||𝑆↑↓⟩,𝑧1𝑆𝑧2||1↓↑⟩=−4||↓↑⟩,𝑆𝑧1𝑆𝑧2||1↓↓⟩=4||↓↓⟩.(9) Using these eigenstates, the singlet and triplet states are defined as follows:||1𝑆⟩=√2||||,||𝑡↑↓⟩−↓↑⟩1||||𝑡⟩=↑↑⟩,01⟩=√2||||,||𝑡↑↓⟩+↓↑⟩−1||⟩=↓↓⟩.(10) In this [base kets], the matrix representation of Hamiltonian is as follows:⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝−3𝐻=4𝐷𝐽−2√2𝐷0−2√2−𝐷2√2𝐽4−√22√ℎ00−22â„Žğ½4−√22â„Žâˆ’ğ·2√2√0−22â„Žğ½4⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠.(11) We diagonalize the Hamiltonian (11) and determine the GS of the systems as follows: ||||||𝑡𝐺𝑆⟩=𝐴𝑆⟩+𝐵1||𝑡⟩+−1⟩||𝑡+𝐶0⟩.(12) In which 𝐴, 𝐵, and 𝐶 are as follows: 𝐽𝐴=4𝐷,𝐵=2√2√,𝐶=2ℎ2.(13) In order to define the characteristics of the GS of the model in different subspaces of the GS phase diagram, we first calculated the magnetic order parameter 𝑀𝑥,𝑦,𝑧 and 𝑀𝑥,𝑦,𝑧𝑠𝑡as follows:𝑀𝑥,𝑦,𝑧||1=⟨GS𝑁𝑛𝑆𝑛𝑥,𝑦,𝑧||𝑀GS⟩,𝑥,𝑦,𝑧𝑠𝑡||1=⟨GS𝑁𝑛(−1)𝑛𝑆𝑛𝑥,𝑦,𝑧||GS⟩.(14) The averaging is calculated on GS. The magnetization along the applied field is plotted via ℎ for different DM vectors in Figure 1. It can be seen that for small 𝐷=0.1, the magnetization for small fields remains close to zero. Increasing ℎ up to ℎ>â„Žğ‘, the magnetization becomes saturated. Due to quantum fluctuations in the presence of 𝐷, there is not any sharp transient in saturation point. For greater values of 𝐷, the magnetization increases by imposing magnetic field. In XXZ model, in ℎ<â„Žğ‘ magnetization has some fluctuations and for ℎ>â„Žğ‘ will be saturated. Imposing transverse external magnetic field, causes removing quantum fluctuation and the spins completely direct along the field, but the magnetization saturation cannot be seen because of broken symmetry in finite systems.

We show that in previous section that by imposing magnetic field ℎ in the presence of DM interaction a staggered magnetization will be created normal to âƒ—âƒ—â„Žğ·- surface using the definition of staggered magnetization we can obtain the following: 𝑀𝑧𝑠𝑡=12||𝑆⟨𝐺𝑆𝑧1−𝑆𝑧2||𝐺𝑆⟩.(15) The staggered magnetization via ℎ for 𝐷=0.1,0.5,1.0. is plotted in Figure 2. Figure 2 shows that applying homogeneous ℎ, creates Neel order in 𝑧 direction. That shows spin-flop ordering. There is a maximum for AM about 0.3. This is the same maximum of an independent dimer which weakly depends on 𝐷. Nersesyan et al. [18] shows that in one-dimensional spiral model by the interaction of second nearest neighbored which has a simple surface, a new phase is created due to breaking parity symmetry. This calls chiral phase and its order parameter is defined as follows: 𝜒𝛼||𝑆=⟨𝐺𝑆1×𝑆2𝛼||𝐺𝑆⟩,(16) where 𝛼 refers to 𝑥-, 𝑦-, 𝑧-axes. There are two types of gapped and gapless energy spectrum chiral phase [19–21].

For studying the GS phase diagram of the spin chain in the external magnetic field and DM interaction, we calculate the chiral order parameter as follows: 𝜒𝑦||𝑆=⟨𝐺𝑆1×𝑆2𝑦||||𝑆𝐺𝑆⟩=⟨𝐺𝑆𝑧1𝑆𝑥2−𝑆𝑥1𝑆𝑧2||𝐺𝑆⟩.(17) In Figure 3, this chiral is plotted as a function of ℎ for different values of 𝐷=0.1,0.5,1.0. As it can be seen, in the absence of DM interaction, the chiral order parameter does not exist in the ground state of the system. When external magnetic field applies, the order parameter changes and starts to increase by increasing the magnetic field. By continuing increase the magnetic field ℎ, chiral order parameter will decrease and for sufficiently values of it will be disappeared. This is a state at which the system is in the saturated magnetization phase and the results are in agreement with analytical results of Section 2.

4. Three Particle Spin-(1/2) Systems

Consider a three-particle spin-(1/2) system whose Hamiltonian is as follows:𝑆𝐻=𝐽1⋅𝑆2+𝑆2⋅𝑆3𝑆+𝐷1×𝑆2+𝑆2×𝑆3î‚î€·ğ‘†âˆ’â„Žğ‘¥1+𝑆𝑥2+𝑆𝑥3.(18) In order to solve the problem, we choose the [base kets] of the Hilbert space as eigenstates of the operator 𝑆𝑧1𝑆𝑧2𝑆𝑧3 as follows that are𝑆𝑧1𝑆𝑧2𝑆𝑧3||1↑↑↑⟩=8||𝑆↑↑↑⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↑↑↓⟩=−8||𝑆↑↑↓⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↑↓↑⟩=−8||𝑆↑↓↑⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↓↑↑⟩=−8||𝑆↓↑↑⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↑↓↓⟩=8||𝑆↑↓↓⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↓↓↑⟩=8||𝑆↓↓↑⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↓↑↓⟩=8||𝑆↓↑↓⟩,𝑧1𝑆𝑧2𝑆𝑧3||1↓↓↓⟩=−8||↓↓↓⟩.(19) In these states, the Hamiltonian matrix isâŽ›âŽœâŽœâŽœâŽœâŽœâŽœâŽâŽžâŽŸâŽŸâŽŸâŽŸâŽŸâŽŸâŽ ğ»=𝐴𝐵𝐶0𝐵000𝐵0𝐴𝐹0𝐵00𝐶0−𝐴𝐶𝐴0𝐶00𝐹000𝐴0𝐶𝐵0𝐴0−𝐴𝐵𝐹00𝐵0𝐴𝐵0𝐴𝐵00𝐶0𝐹00𝐶000𝐶0𝐵𝐶𝐴.(20) In which 𝐽𝐴=4𝐷⟶𝐵=4−ℎ2𝐷⟶𝐶=−2−ℎ2â„ŽâŸ¶ğ¹=−2.(21) We diagonalzed the Hamiltonian and found and the ground state of the system. Then order parameters as follows:𝑀𝑧𝑠𝑡=12||𝑆𝐺𝑆𝑥1+𝑆𝑥2+𝑆𝑥3||,𝑀𝐺𝑆𝑧𝑠𝑡=12||𝑆𝐺𝑆𝑧1−𝑆𝑧2+𝑆𝑧3||,𝜒𝐺𝑆𝑦||𝑆=⟨𝐺𝑆1×𝑆2𝑦+𝑆2×𝑆3𝑦||𝐺𝑆⟩,(22)are calculated for this system, numerically that are shown in Figures 4 and 5. In the three spin case, we find the same behavior which we do not show. Specially three-particle spin-(1/2) system has exactly the same results of two particle system due to defining the order parameters in the particle number unit. In the XXZ model ℎ<â„Žğ‘, the magnetization has some of the quantum fluctuation and the system will be in the magnetic field direction but the magnetization cannot be seen due to the symmetry breakdown in the finite systems.

5. Conclusion

In summary, we have studied Heisenberg model with added Dzyaloshinskii-Moriya interaction in the presence of uniform transverse magnetic field. To this end, we have first implemented a uniform rotation about 𝑥-axis in order to gauge away the added Dzyaloshinskii-Moriya interaction and produce an XXZ model with uniform magnetic and a staggered field normal to âƒ—âƒ—â„Žğ·- surface. We have chosen blocks of two and three spins to be able to solve the model exactly and have also shown how this scheme has capability to capture the right physics. We plotted magnetization, staggered magnetization, and chirality versus magnetic for different DM interactions strength. In the absence of DM interactions there is a sharp change from zero magnetization at ℎ<â„Žğ‘ to saturated region ℎ>â„Žğ‘, [but increasing DM interaction favor to produce a nonzero magnetization at ℎ<â„Žğ‘ region] and there is no sharp transient any more. Staggered magnetization perpendicular to the plan consists of applied magnetic field and DM vector shows spin-flop ordering. The physical systems which prefer spiral chiral configuration usually have competing interactions. For example, when the nearest neighbor (NN) exchange interaction is weak, the next nearest neighbor (NNN) interaction or even Dzyaloshinskii-Moriya (DM) interaction becomes relatively significant. So, we measure chirality in our model versus magnetic field. Our calculations show that in the absence of DM interaction, the chiral order parameter is not in the ground state of the system. But by turning external magnetic field on, the order parameter will be changed and will be increased by increasing the magnetic field. Continuing the increase of magnetic decreases the chiral order and for sufficient values of it will disappear.


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Copyright © 2011 M. R. Soltani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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