Quantum phase transition and magnetic plateau in three-leg antiferromagnetic Heisenberg spin ladder with unequal J1–J2–J1 legs

Journal of Magnetism and Magnetic Materials 397 (2016) 319–324

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Quantum phase transition and magnetic plateau in three-leg antiferromagnetic Heisenberg spin ladder with unequal J1–J2–J1 legs Shuling Wang a,n, Sicong Zhu b, Yun Ni c, Li Peng b, Ruixue Li d, Kailun Yao b a

School of Science, Hebei University of Engineering, Handan 056038, China School of Physics and Wuhan National High Magnetic Field Center, Huazhong University of Science and Technology, Wuhan 430074, China c Wenhua College, Huazhong University of Science and Technology, Wuhan 430074, China d School of Science, Henan Institute of Engineering, Zhengzhou 451191, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 29 March 2015 Received in revised form 9 July 2015 Accepted 29 August 2015 Available online 1 September 2015

Magnetic properties of spin-1/2 antiferromagnetic three-leg Heisenberg ladders, where antiferromagnetic interactions in legs are J1, J2 and J1 respectively and in the rungs are J⊥, have been investigated by bond-mean field method. As J⊥ changes, magnetization curves show different behavior. For J⊥ ¼0.5, there are cusps in magnetization curves, while for J⊥ ¼ 3.0, the 1/3 magnetization plateau appears, which can be explained by energy spectra. Furthermore, for J⊥ ¼3.0 the 1/3 magnetization plateaus will become wider or narrow down with J2 changing. In addition, the mean-field bond parameters and the concurrences, which confirm the phase transitions, are also studied. & 2015 Elsevier B.V. All rights reserved.

Keywords: Spin ladder Magnetic plateau Energy spectra Phase transitions

1. Introduction Low dimensional magnetic systems have attracted much attention due to their various fascinating magnetic properties and thermodynamic behavior in the past decades [1–5]. Particularly, spin chains and spin ladders which are typical systems have been investigated theoretically and experimentally [1,2,6–16], and it has been found that the ladders with even legs have gapped low-energy excitations while odd-leg ladders are gapless [3,4,7]. Much effort has been made to study the spin-1/2 Heisenberg ladders, which have interesting phenomena related to magnetization plateau, quantum critical properties and spin excitation behavior [2,4,6–8,11,12,17–21]. Herein, one of particular interests is the magnetic field induced quantum phase transitions (QPTs). For two leg spin ladders, investigations on both strong-rung spin ladders (C5H12N)2CuBr4 ((Hpip)2CuBr4) [17] and strong-leg spin ladders (C7H10N)2CuBr4 (DIMPY) [12] under different magnetic field have furthered the understanding of their phase diagram. Besides, the quantum criticality in the three-leg ladders have been also studied widely, by bond-mean-field theory (BMFT), density-matrix renormalization group (DMRG) technique, and the strong-coupling series expansion [2,4,18–20,22]. In 2008, Azzouz et al. calculated n

Corresponding author. E-mail address: [email protected] (S. Wang).

http://dx.doi.org/10.1016/j.jmmm.2015.08.119 0304-8853/& 2015 Elsevier B.V. All rights reserved.

the magnetic field dependence of bond parameters and magnetization in S¼1/2 three-leg antiferromagnetic Heisenberg ladders, and found that the plateau at one third of the saturation magnetization appears only when the rung-to-leg coupling exceeds a threshold value [18]. Moreover, the Spin-Peierls instability in the three-leg ladder coupled to phonons has also been studied [20–22]. Numerical results showed that the leg-dimerizations antiferromagnetic three-leg spin ladders have column and staggered dimerized patterns, and the column dimerized presented lower zero temperature energies [20]. In 2014, the three-leg ladders with leg- and rung-dimerizations have also been discussed [8]. However, studies on spin ladders with unequal legs are lacked, except several investigations about the asymmetric zigzag ladders [23,24]. Here, we concentrate on the three-leg spin ladders with unequal J1–J2–J1 legs, as shown in Fig. 1. Our goal is to study the magnetic process of ladders with unequal legs, and obtain the field-induced multiple-point quantum criticality. This paper is organized as follows: In Section 2, we study the three-leg ladder in a uniform magnetic field using the bond mean-field theory (BMFT). In Section 3, the field dependence of magnetization and the mean-field bond parameters for different rung interaction are investigated. To explain the distinct features of the magnetic curves, we also plot the energy spectrum. Moreover, in order to confirm the appearance of the quantum phase transition, the entanglements are presented. In Section 4, conclusions are drawn.

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S. Wang et al. / Journal of Magnetism and Magnetic Materials 397 (2016) 319–324

2. Model and method The Hamiltonian for the three-leg spin ladders with different couplings along the legs is N ⎧

2

⎪

⎫ ⎪

∑ ⎨ J1 S i,1S i + 1,1 + J2 S i,2 S i + 1,2 + J1 S i,3 S i + 1,3 + ∑ J⊥ S i, j S i, j + 1⎬

H=

⎪ i ⎩

j =1 N

⎪ ⎭

3

− gμB B ∑ ∑ S iz, j i

(1)

j

where the index i¼ 0, 1,…, N 1 labels the N sites on each of the legs and j¼1, 2, and 3 labels the legs. The periodic boundary conditions are chosen along the leg direction and the open boundary conditions along the rung direction. J1 40, J2 40 and J⊥ 40 are the antiferromagnetic Heisenberg exchange coupling constants along the chains and the rungs, respectively. For convenience, we define the reduced magnetic field h¼gmBB. Using the two-dimensional generalized Jordan–Wigner transformation [25], the spin operators at site of the three legs are written as follows: i−1

Si−,1 = ci,1eiϕi,1, ϕi,1 = π = ci,2

eiϕi,2 ,

ϕi,2 = ϕi,1 + πni,1

Si−,3 = ci,3 eiϕi,3 , ϕi,3 = ϕi,2 + πni,2 1 2

Siz, j = ci+, jci, j −

(2)

where ni, j = ci+, jci, j is the occupation operator for the JW fermions. The hopping terms of Eq. (1) are written using the BMFT, which approximates the sum of the phase differences by π, 0,π, 0 along the legs [18,25,26]. After the JW transformation, the Hamiltonian (1) becomes

H=

∑ J1 [ 21 (c2+i,1c2i + 1,1eiπ + c2+i − 1,1c2i,1 + h. c. ) + 1 1 − 2 )(c2+i + 1,1c2i + 1,1 − 2 ) 1 1 (c2+i − 1,1c2i − 1,1 − 2 )(c2+i,1c2i,1 − 2 )]

−

+ +

1 J1 [ 2 (c2+i,3c2i + 1,3 eiπ

−

+

−

+ h. c . ) +

1 1 (c2+i,3c2i,3 − 2 )(c2+i + 1,3c2i + 1,3 − 2 ) 1 1 + (c2+i − 1,3c2i − 1,3 − 2 )(c2+i,3c2i,3 − 2 )]

+

1 J⊥ [ 2 (c2+i,1c2i,2

+

c2+i − 1,2c2i − 1,3

+

(c2+i − 1,1c2i − 1,1

+

(c2+i,1c2i,1

+

(c2+i − 1,2c2i − 1,2

+

(c2+i,2c2i,2

−

−

+

c2+i − 1,1c2i − 1,2

1

1 N

−

1 ) 2

− 2 )(c2+i − 1,3c2i − 1,3 − 2 ) −

(6)

(7)

i 2πN

∑ eik (i − j) ∫ 1 , kB T

dω [g (k, ω + iο+) − g (k, ω − iο+)] e βω + 1

(8)

kB and T are the Boltzman's constant and the ab-

solute temperature, respectively. The concurrence of the bipartite quantum state characterizing the quantum entanglement, can be calculated as

1

1

1 4

∑ g (k) eik (i − j)

where β =

C = max {0, λ1 − λ2 − λ3 − λ 4 }

1 )] 2

1

− gμB B ∑ ∑ (ci+, jci, j − 2 ) j=1

+ ⟨ci+, jci, j ⟩ci++ 1, jci + 1, j − ⟨ci++ 1, jci + 1, j ⟩⟨ci+, jci, j ⟩

k

+ h. c . )

1 )(c2+i,3c2i,3 2

+ ⟨ci, j ci++ 1, j⟩⟨ci + 1, j ci+, j⟩ + ⟨ci++ 1, jci + 1, j ⟩ci+, jci, j

⟨b+j ai ⟩ =

c2+i,2c2i,3

1 − 2 )(c2+i − 1,2c2i − 1,2 1 1 )(c2+i,2c2i,2 − 2 ) 2

3 i

+

⟨ci, j ci++ 1, j⟩ci+, jci + 1, j + ⟨ci + 1, j ci+, j⟩ci++ 1, jci, j

The integral of the wave vector k is along the chain direction. So, the momentum space Green's function g (k, ω)can be characterized as a function of wave vector k and the elementary excitation spectrum ω = ω (k ). According to the standard spectral theorem, the correlation function of the fermion operators can be obtained by,

1 )] 2

c2+i − 1,3c2i,3

⎛ + 1 ⎞⎛ 1⎞ ⎜ ci, jci, j − ⎟ ⎜ ci++ 1, jci + 1, j − ⎟ = ⎝ 2 ⎠⎝ 2⎠

k

1 ) 2

−

1 )(c2+i,2c2i,2 2

leg-2, P = ci, j ci+, j + 1 is along the rungs [18,25,26],

Gij =

1

(c2+i − 1,2c2i − 1,2

(5)

For further Fourier transformation into momentum space, the Green's function can be expressed as

+ J2 [ 2 (c2+i − 1,2c2i,2 eiπ + c2+i,2c2i + 1,2 + h. c . ) + 1 )(c2+i + 1,2c2i + 1,2 2

ω ≪ ai ; bj+ ≫ = ⟨[ai ; bj+]+ ⟩ + ≪ [ai , H ]; bj+ ≫

1

(c2+i,1c2i,1

(c2+i,2c2i,2

(4)

where the subscripts i and j label lattice sites. After the time Fourier transformation, the Green's function is put into the equation of motion,

− 2 ci++ 1, jci + 1, j − 2 ci+, jci, j +

i

+

Gij (t − t‵) = ≪ ai (t ); b+j (t‵) ≫ = − iθ (t − t‵)⟨ai b+j + b+j ai ⟩

By using the equation of motion similar as Eq. (5) for the higher-order Green's function ≪[ai , H ] ; b+j ≫ , it will generate the higher-order Green's function, giving rise to an infinite set of coupled equations.The quartic Ising terms are treated by Hartree– Fock approximation using the bond parameters Q 1 = ⟨c2i,1c2+i + 1,1⟩ = ⟨c2i,3 c2+i + 1,3⟩ for leg-1(3) and Q 2 = ⟨c2i,2 c2+i + 1,2⟩ for

3

∑ ∑ n d, f d=0 f =1

Si−,2

Fig. 1. The three-leg ladder with unequal J1–J2–J1 legs. J1, J2 are the antiferromagnetic Heisenberg exchange integral along leg-1(3) and leg-2. J⊥ is the antiferromagnetic Heisenberg exchange integral along the rung.

(3)

We employ the equations of motion method to calculate the retarded Green's function [27,28] for JW fermions, which is described as

(9)

where λ1, λ2, λ3, λ4 are square roots of the eigenvalues of ρ~ij ρij with descending order. ρ~ij = (σiy ⊗ σ jy ) ρij* (σiy ⊗ σ jy ) and σiy is the y component of the Pauli operator. ρ * is the complex conjugation of ij

reduced density matrix ρij [29–31]. Herein, the reduced density matrix ρij are the correlation functions, which can be obtained by Green's function method, and in the conventional basis { ↑↑⟩, ↑↓⟩, ↓↑⟩, ↓↓⟩}, it can take the form

S. Wang et al. / Journal of Magnetism and Magnetic Materials 397 (2016) 319–324

⎛

⎜ i j ⎜

i j ρij = ⎜ ⎜ <σ +P ↑ > ⎜ i j ⎜ <σ +σ +> ⎝ i j

<σi−P ↑j > <σi−σ j−> ⎞ ⎟ <σi−σ j+> <σi−P ↓j >⎟ ⎟ <σi+σ j−> ⎟ ⎟ <σi+P ↓j > ⎟⎠

1 2

z

x

±

(10)

y

where P ↑↓ = ± S and σ = S ± iS . The concurrences of the bipartite quantum state along the leg-1(3), leg-2, and rungs between the leg-1(3) and leg-2 are C12, C34, C13, respectively. The concurrences equal zero corresponding to unentangled states and one corresponding to maximally entangled states. Then, the average magnetization M per cell and magnetization M1, M2 along leg-1(3) and leg-2, can be described as

M=

1 3N

∑ ⟨Siz, j ⟩, i, j

M1 (3) =

1 N

∑ ⟨Siz,1(3) ⟩, i

M2 =

1 N

∑ ⟨Siz,2 ⟩ i

(11)

These equations can be calculated self-consistently, and the magnetizations, the concurrences and energy spectra can be obtained. In the following discussions, we set J1 ¼1.0 as the energy unit. For J⊥, we can find that there are two regimes in the field dependence of magnetization. For simplification of analysis, we only concentrate on the following two cases: (1) small rung coupling, we set J⊥ ¼ 0.5; (2) large rung coupling, we set J⊥ ¼ 3.0.

3. Results and discussions As shown in Fig.2(a) and (b), the magnetization as a function of field is presented for J⊥ ¼0.5 and J⊥ ¼ 3.0 under different J2. When

321

h¼0.0, the magnetization is zero no matter J⊥ is small or large, which can be confirmed by the energy spectra, shown in Fig.2 (c) and (d). It can be found that for J⊥ ¼0.5 and J⊥ ¼3.0, there is no gap between the energy bands near the Fermi energy. While under intermediate field, three cusps appear for J⊥ ¼0.5, and intermediate magnetization plateaus at one third of the saturation magnetization (Ms/3) occur for J⊥ ¼ 3.0, which is also found in the three-leg ladders with equal legs and strong rung coupling [18]. However, the width of the one third of saturation magnetization plateau we calculated is about 3 J for J1 ¼J2 ¼1.0, which is larger than the numerical results(about 1.88 J) of Tandon et al. [19]. When J1 is not equal to J2, using density-matrix renormalizationgroup method the one third of saturation magnetization plateau may appear for strong rung coupling, but the width of the Ms/3 plateau may be different from our results. From Fig. 2(d), it can be seen that the gap between the energy bands ω1 and ω2 contributes to the Ms/3 plateau for J⊥ ¼3.0. Fig. 2(b) plots the Ms/3 plateaus for different J2. The results show that when J2 is larger than 1.0, the Ms/3 plateau shrinks and the critical field h3 increases with J2 ascending. As J2 is smaller than 1.0, the critical fields h2 and h3 exhibit complex behavior when J2 changes from 0 to 1.0. The critical field h3 is nearly unchanged when J2 changes from 0.0 to 0.5, and increases when J2 changes from 0.5 to 1.0. The field dependence of magnetization around critical field h2 for J2 ¼ 0.0, 0.5 and 1.0, are drawn in the inset of Fig. 3(d). The critical field h2 firstly increases when J2 changes from 0.0 to 0.5, and then decreases from J2 ¼0.5 to 1.0, which would be explained by the band structure. To illuminate the magnetization curves for different J2 under J⊥ ¼3.0, the energy spectra for the upper three energy bands are

Fig. 2. Zero-temperature magnetization as a function of field for (a) J1 ¼1.0, J⊥ ¼0.5, J2 ¼ 0.0, 0.1, 0.5, 1.0, 1.5, 2.0 and 3.0, respectively; (b) J1 ¼ 1.0, J⊥ ¼ 3.0, J2 ¼0.0, 0.5, 1.0, 1.5, 2.0, 3.0 and 4.0, respectively. Energy spectra at h ¼ 0.0 for J1 ¼ 1.0, J2 ¼1.5, (c) J⊥ ¼ 0.5; (d) J⊥ ¼3.0. h1, h2, h3 are the critical fields. The horizontal dashed line designates the Fermi level at h ¼0.0. ω1, ω2, and ω3 are the energy bands above Fermi energy.

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S. Wang et al. / Journal of Magnetism and Magnetic Materials 397 (2016) 319–324

Fig. 3. Energy spectra for the upper three energy bands of three-leg spin ladders for J1 ¼1.0, J⊥ ¼ 3.0, at h ¼ 0.0, (a) for J2 ¼ 1.0, 2.0, 3.0; and (c) J2 ¼ 0.0, 0.5, and 1.0; (b) at h ¼3.0 for J2 ¼ 2.0, 3.0; (d) at h ¼ 4.5, for J2 ¼ 0.5, 1.0. The inset of Fig. 3(d) is zero-temperature magnetization as a function of field for J1 ¼ 1.0, J⊥ ¼3.0, J2 ¼ 0.0, 0.5 and 1.0 around the critical field h2.The magenta dot lines in (a) and (b) are Fermi levels.

Fig. 4. The magnetization curves (M1, M2) of the sites in leg-1(3) and leg-2, and for J1 ¼1.0, J2 ¼1.0 and 2.0, respectively, (a) J⊥ ¼ 0.5; (b) J⊥ ¼ 3.0. For comparison, the average magnetizations (M) of unit cell in three-leg ladders are also included.

S. Wang et al. / Journal of Magnetism and Magnetic Materials 397 (2016) 319–324

demonstrated in Fig. 3, and the three energy bands are labeled by ω1, ω2, and ω3 from the bottom up, as shown in Fig. 2(c). For J2 41.0, the width of energy band ω2 is wider as J2 getting greater no matter the field is 0 or not, as shown in Fig. 3(a) and (b). In addition, the main role of the magnetic field is to lift the Fermi level. Consequently, the gap between the energy bands ω1 and ω2 for J2 ¼3.0 is smaller than that for J2 ¼ 2.0 with field or without field, implying that the Ms/3 shrinks with J2 increasing. The critical field h3, signaling the beginning of the saturation plateau, denotes the open of the pseudogap. From Fig. 3(b), it can be found that the top of the energy band ω2 (ω3) is higher for J2 ¼3.0 than that for J2 ¼2.0, which confirms that open the pseudogap needs more energy with J2 increasing, and explains the increasing of critical field h3, as shown in Fig. 2(b). For J2 o 1.0, it can be found that the width of the energy band ω2 (ω3) decreases when J2 changes from 0.5 to 1.0 without field. However, when the system is under field, the width of the energy band ω2 (ω3) increases when J2 changes from 0.5 to 1.0, which is shown in Fig. 3(c) and (d). Therefore, when J2 changes from 0.5 to 1.0, the magnetic field not only lifts the Fermi level, but also plays an important role on the shape of the energy bands. Moreover, the influence of field on shape of energy bands is more important than that on the Fermi level. For J2 ¼0.0, the field only brings the energy band down, similar to that for J2 41.0. When J2 changes from 0 to 0.5, the influence of field on the width of the energy band ω2 for J2 ¼0.5 make the energy band ω2 close to Fermi level need more energy, comparing with that for J2 ¼0.0. Correspondingly, the critical field h2, signaling the close of energy gap, becomes greater when J2 changes from 0.0 to 0.5, as shown in the inset of Fig. 3(d).

323

Fig. 4 plots magnetization for the sites in different legs, M1 for leg-1(3) and M2 for leg-2. As shown in Fig. 4(a), when J⊥ is smaller, it can be found that there are three cusps in the magnetic curve. When 0 oho h1, M1 increases from 0 and M2 keeps zero. At h1 oho h2, M1 increases rapidly to the saturation magnetization, while M2 ascends more slowly from zero comparing with M1. At h2 oho h3, M2 increases to the saturation value and M1 remains unchanged. The results show that the magnetization M for the three-leg ladder, the unsynchronized behavior of M1 and M2 contributes to the three cusps of the magnetization M. In addition, M2 changes from zero to saturation magnetization more slowly for J2 ¼ 2.0 than that for J2 ¼ 1.0. As shown in Fig. 4(b), when J⊥ is larger (J⊥ ¼3.0), there are also plateaus in the magnetization curve for M1 and M2, which is similar to the magnetization behavior of M for unit cell. When 0 oho h1, M1 increases, while M2 decreases from zero. At h1 oh oh2, both M1 and M2 keep unchanged. Both M1 and M2 increase at h2 ohoh3, then reach the saturation magnetization at h3. The bond parameters and concurrences are good indicators of quantum phase transitions, which can be used to characterize quantum correlation. To further explain the magnetization behavior of 3-leg ladders, the correlations and concurrences for two nearest sites along legs and rungs are plotted in Fig. 5. Fig. 5(a) and (c) present the magnetic field dependence of the mean-field bond parameters and concurrences for small J⊥ (J⊥ ¼ 0.5). At 0 ohoh1, the bond parameter Q1 decreases as h increases, while the bond parameters Q2 and P almost remain unchanged, as a result the magnetization along leg-1(3) increases and M2 remains almost unchanged with magnetic field ascending as shown in Fig. 4(a).

Fig. 5. Zero-temperature mean-field bond parameters as a function of field for J1 ¼ 1.0, J2 ¼ 2.0, (a) J⊥ ¼0.5; (b) J⊥ ¼3.0. Concurrences between two spins along leg-1(C12), leg-2(C34) and rung between leg-1 and leg-2(C13) as a function of field for J1 ¼1.0, J2 ¼ 2.0, (c) J⊥ ¼ 0.5; (d) J⊥ ¼ 3.0. Zero-temperature mean-field bond parameters and concurrences C12, C34, and C13 for J2 ¼1.0 are included for comparison.

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S. Wang et al. / Journal of Magnetism and Magnetic Materials 397 (2016) 319–324

As h continues to increase, the three bond parameters (Q1, Q2 and P) all decrease, and Q1 becomes zero when the field reach h2, reflecting that the correlation between the two nearest sites along leg-1(3) decreases more rapidly than that along leg-2, which contribute to the more rapid increase of M1 between the field region h1 ohoh2. When the field exceeds h3, all the three parameters vanish, and all the sites are uncorrelated, which is confirmed by the concurrences characterizing the quantum entanglements of two sites. As shown in Fig. 5(c), when field is larger than h3, all the concurrences become zero, reflecting the spins of all the sites become parallel. Moreover, we also find that the magnetic field region for J2 ¼2.0 between critical field h2 and h3 enlarges in contrast with that for J2 ¼1.0, which can be used to explain the slowly increase of the M2 under J2 ¼2.0. Fig. 5(b) and (d) shows the bond parameters and concurrences for large J⊥ (J⊥ ¼3.0). Q1 and Q2 decrease as h increases at 0 ohoh1, and then the bond parameters are all characterized by a plateau in the field region (h1 o hoh2). When h2 o hoh3, Q1 first decreases and then increases slowly to zero, while Q2 first increases and then decreases. For the bond parameter P, it keeps unchanged at 0 ohoh2, then decreases to zero for h2 ohoh3. From Fig. 5, the results for the bond parameters and concurrences describe that the cusps correspond to the critical fields in magnetization curves well.

4. Conclusions We have studied the spin-1/2 antiferromagnetic three-leg Heisenberg ladder with unequal J1–J2–J1 legs under external magnetic field by the bond mean-field theory. For small rung coupling J⊥, only cusps show up in the magnetic curve. However, for large rung coupling J⊥, the one third of saturate magnetization plateau occurs, and the critical fields characterizing the phase transitions perform different behavior under J2 41.0 and J2 o1.0. This is because that for J2 o1.0, the magnetic fields not only make a contribution to lift the Fermi energy, but also have an important effect on the shape of the energy bands. While for J2 41.0, comparing with the effect on the shape of the energy bands, magnetic fields plays dominate role on the Fermi level. Also, the magnetization of sites in different legs are discussed, and unsynchronized increase of magnetization in leg-1(3) and leg-2 contributes to the three cusps in the magnetization of ladders. Moreover, to further understand the phase transitions for different rung coupling J⊥ and J2, the mean-field bond parameters and concurrences under the field are calculated.

Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants no. 11274130, No. 11274128, and No. 11404119, and the Scientific Research Project of Hubei Provincial Education Department under Grants no. B2014248.

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