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Magnetization Plateaus in a Geometrically Frustrated Anisotropic Four-Leg Nanotube
R. Jafari et al
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DOI: 10.22661/AAPPSBL.2019.29.3.48

Magnetization Plateaus in a Geometrically Frustrated Anisotropic Four-Leg Nanotube

R. JAFARI,1, 2, * SAEED MAHDAVIFAR,3 AND ALIREZA AKBARI4, 5, 6, 1
1 DEPARTMENT OF PHYSICS, INSTITUTE FOR ADVANCED STUDIES IN BASIC SCIENCES (IASBS),
ZANJAN 45137-66731, IRAN
2 DEPARTMENT OF PHYSICS, UNIVERSITY OF GOTHENBURG, SE 412 96 GOTHENBURG, SWEDEN
3 DEPARTMENT OF PHYSICS, UNIVERSITY OF GUILAN, 41335-1914, RASHT, IRAN
4 ASIA PACIFIC CENTER FOR THEORETICAL PHYSICS (APCTP), POHANG, GYEONGBUK, 790-784, KOREA
5 DEPARTMENT OF PHYSICS, POSTECH, POHANG, GYEONGBUK 790-784, KOREA
6 MAX PLANCK POSTECH CENTER FOR COMPLEX PHASE MATERIALS, POSTECH, POHANG 790-784, KOREA

*jafari@iasbs.ac.ir; rohollah.jafari@gmail.com

We have studied a frustrated anisotropic four-leg spin-1/2 nanotube using a real space quantum renormalization group (QRG) approach in the thermodynamic limit. We show that in the limit of weakly interacting plaquettes, the model is mapped onto a 1D spin-1/2 XXZ chain in a longitudinal magnetic field under QRG transformation. Analysis of the effective Hamiltonian reveals that the spin nanotube displays both first and second order phase transitions accompanied by fractional magnetization plateaus. We also show that the anisotropy significantly changes the magnetization curve and the location of phase transition points. Moreover, using a numerical exact diagonalization method, the ground state phase diagram was studied. Our numerical results are in complete agreement with the known analytical results.

INTRODUCTION

Frustrated spin systems are known to have many intriguing properties that are different from conventional magnetic systems [1, 2]. Frustration induces unconventional magnetic orders [1-3] or even a disorder [1, 2]. The study of frustrated systems has attracted much more attention with the discovery of J1 - J2 chain materials like CuGeO3 [4], and has been developed by synthesizing of odd number (n) of the legs spin tube, such as [(CuCl2tachH)3 Cl]Cl2 [5] and CsCrF4 [6] with n = 3, and Na2V3O7 [7] with n = 9. Spin tubes with an odd number of legs and only nearest neighbor antiferromagnetic (AFM) intrachain coupling are geometrically frustrated. According to the Lieb-Schultz-Mattis theorem [8], the ground state of such systems is either gapped with a broken translational invariance, or gapless (non- degenerate) [9].

Recently, four-leg spin-1/2 nanotube Cu2Cl4D8C4SO2, with next-nearest neighbor (NNN) AFM interaction, diagonally coupling adjacent legs, has been established as a new frustrated spin tube [10-12] (Fig. 1).

 

Fig. 1: A schematic plot of a frustrated four-leg spin tube. The interaction along the legs is characterized by J1 (red lines) and J0 shows the intra-plaquettes interaction (black lines). The diagonal interaction Jd has been shown by the green lines.

The Hamiltonian of the geometrically frustrated anisotropic four-leg spin tube (FAFST) model in the presence of a magnetic field on a periodic tube of N sites is given by

(1)

where we define

(2)

with n = (0, 1, d). Here J0 > 0, J1 > 0 and Jd > 0 are the plaquette, leg, and diagonal exchange couplings respectively, and the corresponding easy-axis anisotropies are defined by Δ0, Δ1 and Δd. Furthermore, σ = (σ x, σ y, σ z) denotes the Pauli matrices, and h represents a magnetic field to point along the z-direction. Without loss of the generality, we now rescale all the energy parameters in the unit of J0 by considering J0 = 1.

Four leg spin tubes with only nearest neighbor AFM exchange are not frustrated and have been studied [9, 13-17]. Although the magnetic properties of the FAFST model have been investigated in a few papers [9, 18], an understanding of the quantum phases of the FAFST model on a larger scale is still missing [9]. In addition, the phase diagram and universality class of the model in the presence of anisotropy is unknown. Thus, we are motivated to investigate the FAFST model in the presence of a magnetic field, in the strong plaquette coupling limits, using the real space renormalization group (RSQRG) approach. We show that, in the strong plaquette coupling limit, under RSQRG transformation, the FAFST model maps onto the one-dimensional (1D) spin-1/2 XXZ model in the presence of an effective magnetic field. We also show that when the leg and frustrating couplings are the same (maximum frustration line), only first order quantum phase transitions are observed at zero temperature. The magnetization per particle process exhibits fractional plateaus at zero, one-quarter, one-half and three-quarter of the saturation magnetization. Away from the maximum frustrating line, the model exhibits both first and second order quantum phase transitions. In addition, we applied the numerical Lanczos method for the finite size spin-1/2 nanotubes and the obtained results fully agreed with the QRG and support the mentioned behaviors.

REAL SPACE QUANTUM RENORMALIZATION GROUP

Real space quantum renormalization group (RSQRG) method can be chosen to study lattice systems when dealing with zero temperature properties of many-body systems with a large number of strongly correlated degrees of freedom [19-22]. Application of the RSQRG on lattice systems implies the construction of a new smaller system corresponding to the original one with new (renormalized) interactions between the degrees of freedom [23-25].

In this paper we have implemented Kadanoff's block method to study FAFST. In Kadanoff's method, the lattice is divided into disconnected blocks of nB sites each where the Hamiltonian is exactly diagonalized. This partition of the lattice into blocks induces a decomposition of the Hamiltonian 𝓗 into an intrablock Hamiltonian 𝓗B and an interblock Hamiltonian 𝓗BB where the block Hamiltonian 𝓗B is the sum of the commuting terms , each acting on the Ith block of the chain. Each block is treated independently to build the projection operator P0 onto the lower energy subspace. The projection of the Hamiltonian is mapped to an effective Hamiltonian 𝓗eff that acts on the renormalized subspace where the interaction between blocks defines the effective interaction of the renormalized chain, where each block is considered as a new single site. The perturbative implementation of this method has been discussed comprehensively [3, 20, 21], and the effective Hamiltonian up to first order corrections is given by

(3)

with

Fig. 2: A schematic plot of the decomposition of a four-leg spin tube into plaquette blocks where each plaquette is replaced by an effective single site under the renormalization process.

RENORMALIZATION OF THE MODEL WITH STRONG PLAQUETTE COUPLING (WEAKLY INTERACTING PLAQUETTES)

To apply the QRG scheme to the model in the strong plaquette coupling limit, we consider the Hamiltonian of Eq. (1), and we split the spin tube into blocks where each contains an independent plaquette (see Fig. 2). The Hilbert space of each plaquette has sixteen states, which consists of two spin-0 singlets, nine spin-1 triplets and five spin-2 quintuplets [18]. The plaquette Hamiltonian and the four lowest eigenvalues of the plaquette Hamiltonian labeled by e0, e1, e2, e3, and their corresponding eigenstates are given in the appendix. Since energy level crossing occurs between the four lowest eigenstates of the block Hamiltonian, the projection operator, P0, can be different depending on the coupling constants. Thus, we specify the regions with the corresponding two lowest eigenvalues to construct their projection operators. We will discuss the phase diagram in terms of the following five different regions, which are classified by the two lowest eigenvalues of the plaquette Hamiltonian.

Region I: e0 as a ground state and e2 as a first excited state

In this region we have h < Δ0 - 1, and to the first order corrections the effective Hamiltonian leads to the 1D exactly solvable transverse field Ising model, i.e.,

(4)

Here

(5)

with

The 1D Ising model in a transverse field is exactly solvable by the Jordan-Wigner transformation [8] and the RSQRG [20]. For simplicity, we consider isotropic interaction on the plaquette Δ0 = 1. Then, the renormalized coupling and transverse field reduces to J′ = 2(Δd - Δ1)/3, and h′ = 1/J′. The phase transition between the paramagnetic and antiferromagnetic/ferromagnetic phases takes place at h′ = 1, under which the system is ferromagnet (Δ1 < Δd) or antiferromagnet (Δ1 > Δd) while the system enters the paramagnetic phase above the critical point h′ > 1. It is remarkable that, by assuming equal anisotropy ratios for the leg and diagonal interactions Δ1 = Δd, the system is always in the paramagnetic phase, where spins align along the direction of the external magnetic field.

Region II: e0 as a ground state and e1 as a first excited state

In the region II, we have

where e0 and e1 are the ground state and the excited state of the plaquette Hamiltonian, respectively. This leads the effective Hamiltonian to the well-known 1D XXZ model in the presence of an external magnetic field, which can be solved exactly by the Bethe Ansatz method [26, 27]

(6)

where the couplings of renormalized Hamiltonian are given by

(7)

Region III: e1 as a ground state and e0 as a first excited state

This region is defined by the situation

where e1 is the ground state and e0 is the excited state of the plaquette Hamiltonian. One can show that the first-order effective Hamiltonian is the same as the former case, Eq. (6), with the negative field, and that the couplings are defined as before in Eq. (7).

Region IV: e1 as a ground state and e3 as a first excited state

The ground state and the first excited state of the plaquette Hamiltonian are e1 and e3 respectively, for the following field:

In this region the effective Hamiltonian is also similar to region II, with different coupling constants defined by

(8)

Region V: e3 as a ground state and e1 as a first excited state

In the field that fulfills h > Δ0 + 1, e3 is the ground state and e1 is the first excited state. In this region the effective Hamiltonian up to the first order is the same as in region IV, with the magnetic field in opposite direction, and the coupling constants are the same as in Eq. (8).

PHASE TRANSITION

As shown, the renormalized Hamiltonian in the strong plaquette coupling limit is different than the original one, i.e. FAFST, when attempting to find the recursion relation. However, the effective Hamiltonians are exactly solvable [26-28] and it enables us to predict distinct features of the spin tube in the strong plaquette coupling limit. To prevent complexities, we restrict our study to the case Δ0 = 1 and h ≥ 0. In such a case, our analysis does not cover the region I and we only consider the regions II-V where the expected FAFST models in the presence of the magnetic field are mapped to the well-known 1D spin 1/2 exactly solvable models.

First order phase tTransition J1 = Jd

In the case of the equal inter-plaquette couplings J1 = Jd, the frustration is maximum, and the effective model reduces to the well-known 1D spin-1/2 Ising model in a longitudinal magnetic field,

(9)

The ground state properties of this model have been investigated using the RSQRG method [21]. This model shows a first order transition from a classical antiferromagnetic ordered phase to a saturated ferromagnetic phase at Δ′ = h′. Depending on the values of the anisotropy parameter Δ′ and the magnetic field h′, the effective Hamiltonian reveals two magnetization (per site) plateaus Mzeff = 0 and Mzeff = ±1/2 corresponding to the antiferromagnetic and the ferromagnetic phases, respectively. Consequently, the first order phase transition points of FAFSTs are given by

(10)

Additionally, the magnetization in FAFSTs are connected to the magnetization plateaus in the effective Hamiltonian using the renormalization equation (see appendix).

For the plateaus at Mzeff = 0, Mzeff = ±1/2 in the curve of magnetization (per site) versus h′ in the effective model translate into plateaus at MzST = 1/8, MzST = 0, and MzST = 1/4 in the curve of magnetization (per site) versus h in the FAFST. There is also a renormalization equation for for linking the magnetization plateaus in the effective Hamiltonian to the magnetization curve in a FAFST (see appendix). In such a case, plateaus at Meff =0, Meff = ±1/2 in the magnetization curve of the effective model turn into MzST = 1/4, MzST = 3/8 and MzST = 1/2 in the magnetization curve of the FAFST model.

 

Fig. 3: Magnetization (per site) of FAFST versus the magnetic field h obtained by QRG transformation on the maximum frustration line J1 = Jd, with couplings and anisotropic parameters: (a) Δ1 = Δd = 0.5 and two cases of Δ0 = 0 and Δ0 = 1, and (b) Δ0 = 1 and three cases of Δ1 = 0; Δd = 1, Δ1 = 1; Δd = 0.5, and Δ1 = 1; Δd = 1. (c) The same quantity obtained by the numerical Lanczos results on finite spin-1/2 nanotube systems with size N = 24, for different values of the exchanges as Δ0 = 0, and J1 = Δ1 = Jdd = 0.5. The inset shows the corresponding energy gap.

The magnetization curves of FAFST along the maximum frustration line (J1 = Jd) have been shown in Figs. 3 (a and b) based on the numerical RSQRG results. To examine the anisotropy effects, the magnetization curves of FAFST have been plotted versus the magnetic field h for different values of anisotropies. Notice that in Fig. 3(a) the magnetization plateaus has been depicted versus h for an isotropic case Δ0 = 1, J1 = Δ1, Jd = Δd (dashed-blue curve), which shows quantitatively excellent agreement with numerical density matrix renormalization group results [18]. This result indicates that the RSQRG is a good approach to study the critical behavior of FAFST in the thermodynamic limit.

As seen in Fig. 3, the location of critical points and the width of the magnetization plateaus are controlled by the anisotropies according to Eq. (10). It is to be noted that the renormalized subspace specified by the singlet (e0) and triplet (e1) states is separate at the level crossing point from the renormalized subspace defined by the triplet (e1) and quintuplet (e3) states. Therefore, for the cases that is greater than hl, the point hC3= hC4 is not the critical point and the level crossing point would be a first order phase transition point. From Fig. 3(b) one can clearly see that the width of Mz(ST) =1/8 and Mz(ST) = 3/8 plateaus reduced by decreasing the inter-plaquette anisotropies (Δ1 + Δd).

To accomplishment of our investigation, using the numerical Lanczos method, we have studied the effect of an external magnetic field on the ground state magnetic phase diagram of the aforementioned FAFST model. In Fig. 3(c), we have presented our numerical results. In this figure on top of the magnetization, in the inset we plot the energy gap as a function of the magnetic field for a tube size N = 24 and different values of the exchanges according to the Δ0 = 0, J1 = Δ1 = 0.5 and Jd = Δd = 0.5. As is seen, in the absence of the magnetic field the FAFST model is gapped. By increasing the magnetic field, the energy gap decreases linearly and vanishes at the first critical field. By ever increasing the magnetic field, the energy gap closes in three different and critical magnetic fields, independent of the system size. After the fourth critical field, the gap opens again and a sufficiently large field becomes proportional to the magnetic field, which is known as an indication of the paramagnetic phase. On the other hand, the magnetization is zero in the absence of the magnetic field at zero temperature. By increasing the magnetic field, besides the zero and saturation plateaus, three magnetization plateaus at M = 1/8, M = 2/8, M = 3/8 are observed. We have to mention that the critical fields estimated by the numerical Lanczos method are in complete agreement with our analytical results presented in Fig. 3(c).

Fig. 4: Phase diagram: h vs (Δ1d) along the line J1 = J2 obtained by QRG transformation for Δ0 = 1.

The magnetic phases of FAFST along maximum frustration line J1 = Jd has been shown versus (Δ1 + Δd) and h in Fig. 4 for Δ0 = 1, based on the RSQRG approach. As it can be observed, Mz(ST)= 3/8 and Mz(ST) = 1/2, plateaus width linearly increase with frustrating anisotropy (Δ1 + Δd), while width of Mz(ST)=1/8 plateau initially increases linearly with (Δ1 + Δd), and then at hl point reaches to the constant value.

It would be worth mentioning that although at the isotropic point: Δ0 = 1, J1 = Δ1, Jd = Δd, the critical points of FAFST (Eq. 10) reduces to the critical points expression obtained by low energy effective method [18], but the magnetic phase obtained by QRG method is not the same as that of obtained by the low energy effective method [18]. This discrepancy originates from the presence of level crossing point hl, where the system shows first order first transition. The low energy effective method is incapable of capturing the effect of this level crossing point even away from the maximum frustration line J1 = Jd.

Second order phase transition J1Jd

As we mentioned previously, in the case where interplaquette couplings are not equal J1Jd, the FAFST Hamiltonian maps to the effective Hamiltonian, i.e., the 1D spin-1/2 XXZ chain in the presence of the longitudinal magnetic field. This model is exactly solvable by means of the Bethe Ansatz method. Moreover, the properties of the XXZ model in the presence of a magnetic field have been studied using the QRG method [28]. In this subsection, we study the effective Hamiltonian by combining a Jordan-Wigner transformation [8] with a mean- field approximation [29]. By using renormalization equations, which connects the magnetization of the effective model to that of a FAFST, we can obtain the magnetization of the FAFST.

 

Fig. 5: Magnetization (per site) of FAFST versus the magnetic field h obtained by QRG transformation for J1Jd cases, with couplings and anisotropic parameters: (a) Δ0 = 1; J1 = Δ1 = 0.48; Jd = 0.52 and three cases of Δd = (0, 0.52, 1), and (b) Δ0 = 1; J1 = Δ1 = 0.4; Jd = 0.6 and three cases of Δd = (0, 0.6, 1). (c) Numerical Lanczos results of the same quantity on finite spin-1/2 nanotube systems with the system size N = 24. We set Δ0 = 1, and the rest of exchanges are considered for two set of parameters as J1 = Δ1 = 0.48; Jd = Δd = 0.52 (solid-red) and J1 = Δ1 = 0.4; Jd = Δd = 0.6 (dashed-blue).

Hamiltonian maps to the 1D spin-1/2 XXZ chain in the presence of a longitudinal magnetic field. This model is exactly solvable by means of the Bethe Ansatz method. Moreover, the properties of the XXZ model in the presence of a magnetic field have been studied using the QRG method [28]. In this subsection we study the effective Hamiltonian by combining a Jordan-Wigner transformation [8] with a mean- field approximation [29]. Then, by using the renormalization equations, which connects the magnetization of the effective model to that of FAFST, we can obtain the magnetization of FAFST.

To study the effect of anisotropy, the magnetization of FAFST is plotted versus the magnetic field in Fig. 5, for different values of anisotropies. It can be clearly seen that, Mz(ST) = 1/8 and Mz(ST) = 3/8 plateaus width enhances (reduces) by increasing (decreasing) the inter-plaquette anisotropies (Δ1, Δd), and the first order phase transition point at hl fades out gradually by decreasing Δ1 and Δd. As seen, for a small deviations from the maximum frustrated line J1 = 0.48, Jd = 0.52, width of Mz(ST) = 1/8 and Mz(ST) = 3/8, plateaus reduce and jump between plateaus change to smooth curves which is feature of Luttinger liquid phases. As represented in Fig. 5(b), Mz(ST) = 1/8 and Mz(ST) = 3/8 magnetization plateaus are not present for J1 = 0.4, Jd = 0.6. Away from the maximum frustrated line, the magnetization shows only a gapless Luttinger liquid phase, which means that the system consists of decoupled spin-1/2 chains. In other words, the presence of Mz(ST) = 1/8 and Mz(ST) = 3/8 plateaus are very sensitive to frustration.

Again, we have implemented our numerical Lanczos algorithm on the mentioned FAFST model. In Fig. 5(c) we presented our numerical results for different values of the anisotropy parameters. In this figure, the magnetization for a size system N = 24 is plotted as a function of the magnetic field for two set of anisotropy parameters according to Δ0 = 1; J1 = Δ1 = 0.48, Jd = Δd = 0.52 and Δ0 = 1; J1 = Δ1 = 0.4, Jd = Δd = 0.6. As seen in this figure, the place and width of magnetic plateaues are in complete agreement with the analytical results presented in Figs. 5(b and c). One should note that observed oscillations of the magnetization in the Fig. 5(c) have arisen from the level crossings in the finite size systems.

To summarize, for a small deviation from the maximum frustrating line, the one-eight and third-eight magnetization plateaus width, which are sensitive to frustration, can be controlled by the intra-plaquette anisotropies. Away from the maximum frustrating line, where the one-eight and third-eight magnetization plateaus are absent, the intra-plaquette anisotropies can affect the width of a one-quarter magnetization plateau.

SUMMARY

In this paper we have studied the geometrically frustrated anisotropic four-leg spin tube in the absence/presence of magnetic fields by applying the quantum real space quantum renormalization group. We have shown that, in the limit of weakly interacting plaquettes, the FAFST model maps onto the 1D spin-1/2 XXZ model under renormalization transformation. For the case of the same leg and frustrating couplings, maximum frustrating line, the FAFST Hamiltonian reveals only first order quantum phase transitions at zero temperature. In such a case, the FAFST exhibits fractional magnetization plateaus at zero, one-quarter, one-half and three-quarter of the saturation magnetization. We have shown that the magnetization plateaus at one-quarter and three- quarter of the saturation magnetization are very sensitive to frustration and washed out away from the maximum frustrating line. By comparing the remarkable results of the real space renormalization group method to that of the density matrix renormalization group results [9], we see that the real space renormalization group method is a good approach to study the critical behavior of FAFST in the thermodynamic limit.

Acknowledgements: A.A. received financial support through the National Research Foundation (NRF) funded by the Ministry of Science of Korea (Grants No. 2017R1D1A1B03033465, and No. 2019R1H1A 2039733).

Appendix: The plaquette Hamiltonian, the four lowest eigenvalues and their corresponding eigenstates:

The inter-block, 𝓗BB, and intra-block, 𝓗B, Hamiltonians for plaquette decomposition are

and

with eigenstates:

and corresponding eigenvalues are given by

Here |↑>, |↓> are the eigenstates of σz.

The magnetization (per site) in the effective Hamiltonian Mzeff linked to the magnetization (per site) of the spin tube MzST through the renormalization transformation of the σz component of the Pauli matrices. The σz in the effective Hilbert space has the following transformations (α = 1, 2, 3, 4) for each region:

 

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Rouhollah Jafari is a faculty member of the Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran. He received his bachelor's degree in physics from the Iran University of Science and Technology (Tehran, Iran) and his master's degree from IASBS. After obtaining his PhD from IASBS in 2009, he worked as a manager of the research department of Nanosolar system company (Iran), then as a postdoctoral researcher at APCTP (Pohang, Korea) and the University of Gothenburg in Sweden. His research focuses on condensed matter theory and quantum information.

Saeed Mahdavifar is a professor of the department of physics at the University of Guilan, Rasht, Iran. He received his MSc and PhD from the Institute for Advanced Studies in Basic Sciences in Iran. His research focuses on quantum phase transitions in complex low-dimensional quantum magnets.

Alireza Akbari is leading the research group, "Many-body theory and correlated systems", at the Asia Pacific Center for Theoretical Physics (APCTP). He received his PhD from the Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, Iran in 2007. Before joining APCTP, he worked as a scientific researcher at the Max Planck Institute for the Physics of Complex Systems (MPIPKS); Ruhr University Bochum; the Max Planck Institute for Chemical Physics of Solids (MPI-CPfS); and the Max Planck Institute for Solid State Research (MPIFKF).

 
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