DOI: 10.22661/AAPPSBL.2019.29.3.48
Magnetization Plateaus in a Geometrically Frustrated Anisotropic FourLeg Nanotube
R. JAFARI,^{1, 2, }* SAEED MAHDAVIFAR,^{3} AND ALIREZA AKBARI^{4, 5, 6, 1}
^{1} DEPARTMENT OF PHYSICS, INSTITUTE FOR ADVANCED STUDIES IN BASIC SCIENCES (IASBS),
ZANJAN 4513766731, IRAN
^{2} DEPARTMENT OF PHYSICS, UNIVERSITY OF GOTHENBURG, SE 412 96 GOTHENBURG, SWEDEN
^{3} DEPARTMENT OF PHYSICS, UNIVERSITY OF GUILAN, 413351914, RASHT, IRAN
^{4} ASIA PACIFIC CENTER FOR THEORETICAL PHYSICS (APCTP), POHANG, GYEONGBUK, 790784, KOREA
^{5} DEPARTMENT OF PHYSICS, POSTECH, POHANG, GYEONGBUK 790784, KOREA
^{6} MAX PLANCK POSTECH CENTER FOR COMPLEX PHASE MATERIALS, POSTECH, POHANG 790784, KOREA
^{*}jafari@iasbs.ac.ir; rohollah.jafari@gmail.com
We have studied a frustrated anisotropic fourleg spin1/2 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 spin1/2 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 [13] or even a disorder [1, 2]. The study of frustrated systems has attracted much more attention with the discovery of J_{1}  J_{2} chain materials like CuGeO_{3} [4], and has been developed by synthesizing of odd number (n) of the legs spin tube, such as [(CuCl_{2}tachH)_{3} Cl]Cl_{2} [5] and CsCrF_{4} [6] with n = 3, and Na_{2}V_{3}O_{7} [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 LiebSchultzMattis theorem [8], the ground state of such systems is either gapped with a broken translational invariance, or gapless (non degenerate) [9].
Recently, fourleg spin1/2 nanotube Cu_{2}Cl_{4}D_{8}C_{4}SO_{2}, with nextnearest neighbor (NNN) AFM interaction, diagonally coupling adjacent legs, has been established as a new frustrated spin tube [1012] (Fig. 1).
Fig. 1: A schematic plot of a frustrated fourleg spin tube. The interaction along the legs is characterized by J_{1} (red lines) and J_{0} shows the intraplaquettes interaction (black lines). The diagonal interaction J_{d} has been shown by the green lines.
The Hamiltonian of the geometrically frustrated anisotropic fourleg 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 J_{0} > 0, J_{1} > 0 and J_{d} > 0 are the plaquette, leg, and diagonal exchange couplings respectively, and the corresponding easyaxis anisotropies are defined by Δ_{0}, Δ_{1} and Δ_{d}. Furthermore, σ = (σ^{ x}, σ^{ y}, σ^{ z}) denotes the Pauli matrices, and h represents a magnetic field to point along the zdirection. Without loss of the generality, we now rescale all the energy parameters in the unit of J_{0} by considering J_{0} = 1.
Four leg spin tubes with only nearest neighbor AFM exchange are not frustrated and have been studied [9, 1317]. 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 onedimensional (1D) spin1/2 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, onequarter, onehalf and threequarter 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 spin1/2 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 manybody systems with a large number of strongly correlated degrees of freedom [1922]. 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 [2325].
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 n_{B} 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
𝓗_{B} and an interblock Hamiltonian
𝓗^{BB} where the block Hamiltonian
𝓗^{B} 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 P_{0} onto the lower energy subspace. The projection of the Hamiltonian is mapped to an effective Hamiltonian
𝓗^{eff} 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 fourleg 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 spin0 singlets, nine spin1 triplets and five spin2 quintuplets [18]. The plaquette Hamiltonian and the four lowest eigenvalues of the plaquette Hamiltonian labeled by e_{0}, e_{1}, e_{2}, e_{3}, 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, P_{0}, 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: e_{0} as a ground state and e_{2} as a first excited state
In this region we have h < Δ_{0}  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 JordanWigner transformation [8] and the RSQRG [20]. For simplicity, we consider isotropic interaction on the plaquette Δ_{0} = 1. Then, the renormalized coupling and transverse field reduces to J′ = 2(Δ_{d}  Δ_{1})/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 (Δ_{1} < Δ_{d}) or antiferromagnet (Δ_{1} > Δ_{d}) 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 Δ_{1} = Δ_{d}, the system is always in the paramagnetic phase, where spins align along the direction of the external magnetic field.
Region II: e_{0} as a ground state and e_{1} as a first excited state
In the region II, we have
where e_{0} and e_{1} are the ground state and the excited state of the plaquette Hamiltonian, respectively. This leads the effective Hamiltonian to the wellknown 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: e_{1} as a ground state and e_{0} as a first excited state
This region is defined by the situation
where e_{1} is the ground state and e_{0} is the excited state of the plaquette Hamiltonian. One can show that the firstorder 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: e_{1} as a ground state and e_{3} as a first excited state
The ground state and the first excited state of the plaquette Hamiltonian are e_{1} and e_{3} 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: e_{3} as a ground state and e_{1} as a first excited state
In the field that fulfills h > Δ_{0} + 1, e_{3} is the ground state and e_{1} 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 [2628] 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 Δ_{0} = 1 and h ≥ 0. In such a case, our analysis does not cover the region I and we only consider the regions IIV where the expected FAFST models in the presence of the magnetic field are mapped to the wellknown 1D spin 1/2 exactly solvable models.
First order phase tTransition J_{1} = J_{d}
In the case of the equal interplaquette couplings J_{1} = J_{d}, the frustration is maximum, and the effective model reduces to the wellknown 1D spin1/2 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 M_{z}^{eff} = 0 and M_{z}^{eff} = Â±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 M_{z}^{eff} = 0, M_{z}^{eff} = Â±1/2 in the curve of magnetization (per site) versus h′ in the effective model translate into plateaus at M_{z}^{ST} = 1/8, M_{z}^{ST} = 0, and M_{z}^{ST} = 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 M^{eff} =0, M^{eff} = Â±1/2 in the magnetization curve of the effective model turn into M_{z}^{ST} = 1/4, M_{z}^{ST} = 3/8 and M_{z}^{ST} = 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 J_{1} = J_{d}, with couplings and anisotropic parameters: (a) Δ_{1} = Δ_{d} = 0.5 and two cases of Δ_{0} = 0 and Δ_{0} = 1, and (b) Δ_{0} = 1 and three cases of Δ_{1} = 0; Δ_{d} = 1, Δ_{1} = 1; Δ_{d} = 0.5, and Δ_{1} = 1; Δ_{d} = 1. (c) The same quantity obtained by the numerical Lanczos results on finite spin1/2 nanotube systems with size N = 24, for different values of the exchanges as Δ_{0} = 0, and J_{1} = Δ_{1} = J_{d} =Δ_{d} = 0.5. The inset shows the corresponding energy gap.
The magnetization curves of FAFST along the maximum frustration line (J_{1} = J_{d}) 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 Δ_{0} = 1, J_{1} = Δ_{1}, J_{d} = Δ_{d} (dashedblue 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 (e_{0}) and triplet (e_{1}) states is separate at the level crossing point from the renormalized subspace defined by the triplet (e_{1}) and quintuplet (e_{3}) states. Therefore, for the cases that is greater than h_{l}, the point h_{C}_{3}= h_{C}_{4} 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 M_{z}^{(ST) }=1/8 and M_{z}^{(ST) }= 3/8 plateaus reduced by decreasing the interplaquette anisotropies (Δ_{1} + Δ_{d}).
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 Δ_{0} = 0, J_{1} = Δ_{1} = 0.5 and J_{d} = Δ_{d} = 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 (Δ_{1} +Δ_{d}) along the line J_{1} = J_{2} obtained by QRG transformation for Δ_{0} = 1.
The magnetic phases of FAFST along maximum frustration line J_{1} = J_{d} has been shown versus (Δ_{1} + Δ_{d}) and h in Fig. 4 for Δ_{0} = 1, based on the RSQRG approach. As it can be observed, M_{z}^{(ST)}= 3/8 and M_{z}^{(ST)} = 1/2, plateaus width linearly increase with frustrating anisotropy (Δ_{1} + Δ_{d}), while width of M_{z}^{(ST)}=1/8 plateau initially increases linearly with (Δ_{1} + Δ_{d}), and then at h_{l} point reaches to the constant value.
It would be worth mentioning that although at the isotropic point: Δ_{0} = 1, J_{1} = Δ_{1}, J_{d} = Δ_{d}, 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 h_{l}, 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 J_{1} = J_{d}.
Second order phase transition J_{1} ≠ J_{d}
As we mentioned previously, in the case where interplaquette couplings are not equal J_{1} ≠ J_{d}, the FAFST Hamiltonian maps to the effective Hamiltonian, i.e., the 1D spin1/2 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 JordanWigner 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 J_{1} ≠ J_{d} cases, with couplings and anisotropic parameters: (a) Δ_{0} = 1; J_{1} = Δ_{1} = 0.48; J_{d} = 0.52 and three cases of Δ_{d} = (0, 0.52, 1), and (b) Δ_{0} = 1; J_{1} = Δ_{1} = 0.4; J_{d} = 0.6 and three cases of Δ_{d} = (0, 0.6, 1). (c) Numerical Lanczos results of the same quantity on finite spin1/2 nanotube systems with the system size N = 24. We set Δ_{0} = 1, and the rest of exchanges are considered for two set of parameters as J_{1} = Δ_{1} = 0.48; J_{d} = Δ_{d} = 0.52 (solidred) and J_{1} = Δ_{1} = 0.4; J_{d} = Δ_{d} = 0.6 (dashedblue).
Hamiltonian maps to the 1D spin1/2 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 JordanWigner 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, M_{z}^{(ST) }= 1/8 and M_{z}^{(ST) }= 3/8 plateaus width enhances (reduces) by increasing (decreasing) the interplaquette anisotropies (Δ_{1}, Δ_{d}), and the first order phase transition point at h_{l} fades out gradually by decreasing Δ_{1} and Δ_{d}. As seen, for a small deviations from the maximum frustrated line J_{1} = 0.48, J_{d} = 0.52, width of M_{z}^{(ST) }= 1/8 and M_{z}^{(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), M_{z}^{(ST) }= 1/8 and M_{z}^{(ST) }= 3/8 magnetization plateaus are not present for J_{1} = 0.4, J_{d} = 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 spin1/2 chains. In other words, the presence of M_{z}^{(ST) }= 1/8 and M_{z}^{(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 Δ_{0} = 1; J_{1} = Δ_{1} = 0.48, J_{d} = Δ_{d} = 0.52 and Δ_{0} = 1; J_{1} = Δ_{1} = 0.4, J_{d} = Δ_{d} = 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 oneeight and thirdeight magnetization plateaus width, which are sensitive to frustration, can be controlled by the intraplaquette anisotropies. Away from the maximum frustrating line, where the oneeight and thirdeight magnetization plateaus are absent, the intraplaquette anisotropies can affect the width of a onequarter magnetization plateau.
SUMMARY
In this paper we have studied the geometrically frustrated anisotropic fourleg 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 spin1/2 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, onequarter, onehalf and threequarter of the saturation magnetization. We have shown that the magnetization plateaus at onequarter 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 interblock,
𝓗^{BB}, and intrablock,
𝓗^{B}, Hamiltonians for plaquette decomposition are
and
with eigenstates:
and corresponding eigenvalues are given by
Here ↑>, ↓> are the eigenstates of σ^{z}.
The magnetization (per site) in the effective Hamiltonian M_{z}^{eff} linked to the magnetization (per site) of the spin tube M_{z}^{ST} through the renormalization transformation of the σ^{z} component of the Pauli matrices. The σ^{z} in the effective Hilbert space has the following transformations (Î± = 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 lowdimensional quantum magnets. 

Alireza Akbari is leading the research group, "Manybody 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 (MPICPfS); and the Max Planck Institute for Solid State Research (MPIFKF). 
