2 vertically asymmetric diamond chain

Physica A 389 (2010) 5550–5555

Contents lists available at ScienceDirect

Physica A journal homepage: www.elsevier.com/locate/physa

Magnetic frustration induced quantum phase transitions in the S = 1/2 vertically asymmetric diamond chain Yan-Chao Li ∗ State Key Laboratory for Superlattices and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, P.O. Box 912, Beijing 100083, People’s Republic of China

article

info

Article history: Received 12 November 2009 Received in revised form 14 April 2010 Available online 21 August 2010 Keywords: Quantum phase transitions Distorted diamond chain Quantum fidelity Transfer matrix renormalization group

abstract By means of the second derivative of the ground-state energy for a finite-size system method, the quantum phase transitions (QPTs) for the frustrated vertically asymmetric diamond chain (VADC) are investigated. Our results display the plentiful frustration induced quantum phases in the model. Meanwhile, using the transfer matrix renormalization group technique (TMRG), we calculate the fidelity susceptibility and magnetic susceptibility of the VADC model in the thermodynamic limit to give a further understanding of the QPTs. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Quantum phase transitions [1], different from the classical phase transitions caused by thermal fluctuations, are purely driven by quantum fluctuations, such as external magnetic field and press. Studying them can provide a better understanding of properties of natural materials, helping us to exploit and prepare new functional materials, such as high temperature superconductors and novel semiconductors [2–4]. The finding of magnetically mediated superconductivity in a heavyfermion system, for example, shows a tight connection between the quantum phase and properties of materials, which extremely enhanced people’s interest in QPT investigations [4]. Frustration is a purely quantum effect. Its existence can increase quantum fluctuation, resulting in different orders in materials. Therefore, such systems always possess plentiful quantum phases and cause great research interest, such as the compounds SrCu2 (BO3 )2 , Cu3 GeO3 and Cu3 (CO3 )2 (OH)2 [5–8]. Meanwhile, spin frustrated systems with a triangular structure are the most commonly used models, which provide us a simple and effective platform in studying quantum frustration behaviors and QPTs. For instance, the so-called distorted diamond chain model displays plentiful quantum phases for us to understand the competition of interactions among spins [9–12]. In addition, in recent years, a great deal of attention has been paid to QPT study from the perspective of quantum information theory, such as quantum entanglement, geometry phase, and so on. Within such a framework, one important concept in detecting QPTs is that of quantum fidelity (QF), which has been extensively exploited to characterize QPTs [13–20]. It is purely a Hilbert-space geometrical quantity and, comparing with the traditional method, no priori knowledge of order parameter and symmetry breaking is needed. Therefore, it is expected to be a universal tool to investigate QPTs [13,14]. In this paper, we aim to use the fidelity concept as well as other criteria to investigate QPTs of the vertically asymmetric diamond chain (VADC) model with anti-ferromagnetic frustration and give its phase diagram. Due to its triangular-like spin arrangement, as shown in Fig. 1, we expect it has rich physical properties. In Section 2, we first briefly introduce the VADC model and the definition of the quantum fidelity, and then give an outline about the related methods we used. In Section 3,

∗

Tel.: +86 27 87556265. E-mail addresses: [email protected], [email protected].

0378-4371/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2010.08.019

Y.-C. Li / Physica A 389 (2010) 5550–5555

5551

S4,a

S2,a

J1 S1

J3

S5

S3

J2 S2,b

S4,b

Fig. 1. The spin-1/2 vertically asymmetric diamond-chain model. Black dots represent spin-1/2 coupled with exchange constants J1 , J2 , and J3 indicated by the connecting lines.

we present the global phase diagram for the VADC model by using the exact diagonalization technique and TMRG method, analyzing the properties of each phase. Finally, in Section 4, a brief summary will be given. 2. Model and methods The Hamilitonian of the spin-1/2 vertically asymmetric diamond chain can be written as H =

L −

J1 (S2i−1 · S2i,a + S2i,a · S2i+1 ) + J2 (S2i−1 · S2i,b + S2i,b · S2i+1 ) + J3 S2i,a · S2i,b

(1)

i=1

where Si are spin = 1/2 operators at site i, Ji with i = 1, 2, 3 are exchange integrals, and L is the total number of spin repeating cells (the system size is N = 3L). All the coupling constants are supposed to be antiferromagnetic. In the following, we take J2 = 1 as the energy unit. When J1 = J2 , it becomes the diamond chain model and has been investigated in Ref. [21]. However, the J1 ̸= J2 case, which will be studied in this paper, is more general and closer to the actual conditions. The quantum fidelity is defined as the overlap of two ground states with very slightly different parameters [13]. If we assume a general Hamiltonian of a quantum system undergoing a QPT can be written as H (λ) = H0 + λH1 ,

(2)

where H1 is the driving term and parameter λ describes the intensity of a certain kind of interaction, then the quantum ground-state fidelity is defined as F (λ, δ) = |⟨ψ0 (λ)|ψ0 (λ + δ)⟩|,

(3)

where ψ0 is the ground-state wave function and δ is a small variation of λ [13]. It reflects the distinguishability between two ground-state geometrical structures in the Hilbert space and, thereby, can capture the phase transition information whenever a QPT is undergoing. Its second λ derivative is δ independent and sometimes a more effective tool to detect QPTs [17,14]. In addition, it has been proved that the second λ derivative of ground-state energy (SDGE) χg shares the same physical origin with fidelity susceptibility and can be used to identify QPTs. Because only the ground-state energy is needed, the SDGE method is relatively simple to apply. According to the Hellmann–Feynman theorem, the first λ derivative of ground-state energy Eg of H (λ) can be written as

∂ Eg (λ) (4) = ⟨ψg (λ)|H1 |ψg (λ)⟩, ∂λ where ψg is the corresponding wave function of Eg . It is actually the average of driving term Hamiltonian H1 at that state of ψg . Correspondingly, at finite temperature, H1 is a thermal average quantity and can be easy calculated by the TMRG method, which can directly handle infinite spin chains (for an overview, see Ref. [22]). We mark the λ derivative of H1 at a finite temperature as χλ . Since the TMRG method directly deals with infinite-size systems, we expect it could reflect the trace of QPT at a certain finite temperature and avoid the influence of the finite-size effect. 3. Numerical results and discussions We calculate the SDGE χg with respect to J3 of the VADC model by using the exact diagonalization technique with periodic boundary conditions. The χg result and its contour map is plotted in Fig. 2, where N = 18 and δ J3 = 0.01 are taken. From the figure, we can see that dramatic drops divide the figure into three main parts (I), (II), and (III), and curve 1 separates into two curves when J1 > 1.8, resulting in a narrow region (IV). It seems that there exist four different quantum states in the ground state of the VADC model. For a further study, two specific truncations of Fig. 2 under different system sizes N is calculated. Fig. 3(a) shows the truncation at J1 = 1.25 with δ J3 = 0.01, while the region (IV) part at J1 = 2.2 with δ J3 = 0.001 is shown in Fig. 3(b), where 1 and 3 mark the two critical points in curves 1 and 3 of Fig. 2 when J1 = 2.2, respectively. We find that all the drops become sharper and sharper as N increases and would be divergent in the thermodynamic limit. Thus, we conclude that all the drops in Fig. 2 should be regarded as precursors of QPTs.

5552

Y.-C. Li / Physica A 389 (2010) 5550–5555

Fig. 2. (Color online) SDGE χ g with respect to J3 and its contour map of the VADC model for N = 18 on the J1 –J3 plane. The projected lines 1 and 2 of those drops of χ g divide the figure into four parts (I), (II), (III), and (IV).

a

b

Fig. 3. χg under different system sizes N for (a) J1 = 1.25 and δ J3 = 0.01 and (b) J1 = 2.2 and δ J3 = 0.001. All the drops become sharper and sharper as N increases and are expected to be divergent in the thermodynamic limit.

a

b

Fig. 4. Sketch of spin arrangement for two special cases of the VADC model: (a) the J1 = 0 case and (b) the J3 = 0 case. The rectangles indicate equivalent spin centers.

To analyze properties of each phase, we first consider three special cases. When J1 = 0, the spins connected by J3 could be regarded as S = 0 magnetic centers and the remaining spins align in the same direction, then the model becomes an alternating spin-1/2 and spin-0 ferrimagnet (sketched in Fig. 4(a)); when J3 = 0, only the anti-ferromagnetic interactions J1 and J2 have effects. It is a typical alternating spin-1/2 and spin-1 ferrimagnetic chain, as sketched in Fig. 4(b). Therefore, the regions (I) and (II) in Fig. 2 should be ferrimagnetic states; when J1 = J2 , the model changes into the diamond chain model. The SDGE result shows two critical points at around J3 = 0.909 and 2, respectively, which is consistent with the conclusion in Refs. [21,9], where the authors pointed out that two critical points divide the ground state of the diamond model into three phases: the ferrimagnetic phase (ferri) for J3 < 0.909, the tetramer–dimer (TD) singlet phase for 0.909 < J3 < 2 and the dimer–monomer (DM) phase for J3 > 2. Thus, it could be determined that phase (I) is indeed a ferrimagnetic phase,

Y.-C. Li / Physica A 389 (2010) 5550–5555

5553

Fig. 5. (Color online) The lowest energy levels of the VADC model under different Stz subspaces at (a) J1 = 1.25 and N = 12 and (b) J1 = 2.2 and N = 18 (the arrow indicates the degenerate energy-level region).

Fig. 6. The projection of 1E in the J1 –J3 plane for N = 18. The figure gives the global phase diagram: the ferri (light yellow), D (red), SF (purple), and (IV) states. The regions of the ferri and (IV) phases indicated here are consistent with those in Fig. 3; in addition, there is a 1E = 0 line at J1 = 1 when J3 > 2, it is the reflection of the DM phase separating from the ferrimagnetic phase. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

and phase (III) seems to be connected with the TD phase. However, the DM phase seems to be in contradiction with our ferrimagnetic state inference. To clarify the truth, we need to give a further study on the properties of each region. According to Ref. [9], when J1 ̸= J2 , the TD and DM states will smoothly turn to the dimer (D) and spin–flute (SF) states, respectively, and because the four states are all caused by the spin frustration effect, we speculate that there must also be D and SF states for the VADC model. Because the ground state of the ferrimagnetic phase is a N /3 + 1-fold degenerate state with total spin quantum number St = N /6, it is easy to be distinguished from the other states with nondegenerate ground state at finite-size conditions. In addition, although the ground state of both the D and SF states are spin singlet with property St = 0 in the finite-size condition, the lowest excitation is St = 0 for the D state while St = 1 for the SF phase [9]. Therefore, if we set 1E = E1 − E01 , here E1 and E01 are the lowest energy in the Stz = 1 (Stz is the Z -component of the St ) subspace and the first excited state energy in the Stz = 0 subspace, respectively, then we have 1E < 0 for the ferrimagnetic state, 1E = 0 for the SF state, and 1E > 0 for the D state. We first calculate the ground-state energy E under different Stz to study the degenerate conditions as shown in Fig. 5(a), where J1 = 1.25 and N = 12. As it is expected, when J3 < 1.02 and J3 > 2.2, the ground state of the system is a 5-fold degenerate state, i.e., the ferrimagnetic phase, while the middle region is spin gapped. In Fig. 6, we show the projection of 1E for N = 18 in the J1 –J3 plane. The figure gives the global phase diagram: the ferrimagnetic state regions (light yellow) are consistent with those of Fig. 2, while the region (III) is divided into the D (red) and SF (purple) states; meanwhile, a 1E = 0 line at J1 = 1 when J3 > 2 is presented, which is actually the reflection of the DM phase separating from the ferrimagnetic phase. The QPTs results for the J1 = J2 case displayed here are also consistent with those in Ref. [23] and corroborate their findings. In addition, the region (IV) in Fig. 2 is also correspondingly displayed in Fig. 6. Just like the ferrimagnetic state, it also has the property 1E < 0. Does the region relate to the ferrimagnetic state? To answer this question, we show the energy distribution under different Stz subspaces at J3 = 2.2 for N = 18 in Fig. 5(b). The arrow indicates the corresponding

5554

Y.-C. Li / Physica A 389 (2010) 5550–5555

a

b

Fig. 7. (Color online) The J3 derivative of driving term Hamiltonian χJ3 as a function of J3 under different temperatures for (a) J1 = 1.25 and (b) J1 = 2.2. The drops becomes sharper and sharper as temperature decreases and could be regarded as precursors of QPTs at finite temperatures.

Fig. 8. Temperature dependence of χh T at J1 = 0.5 for different J3 . The upturn character at low temperature indicates the ferrimagnetic property of the system.

(IV)-phase region. Different from the ferrimagnetic phase, the phase (IV) is a threefold degenerate state with St = 1, thus it should be a new quantum phase without spin gap even in finite-size systems. To further study the critical behaviors of the VADC model, we investigate the J3 derivative of driving term Hamiltonian χJ3 and magnetic susceptibility χh at finite temperatures by using the TMRG method. Since this method directly deals with infinite systems, we expect the results would be helpful for the judgement of quantum critical behaviors avoiding the finite-size effect. Fig. 7(a) shows the χJ3 as a function of J3 under different temperatures at J1 = 1.25 with interval 1J3 = 0.05. The two dips of χJ3 become sharper and sharper as temperature decreases, and are expected to be divergent in the thermodynamic limit, thus, χJ3 could indeed reflect the quantum critical behaviors at finite temperatures. The two indicated critical precursors are quite close to the SDGE results: one keeps almost the same position at around J3 = 1.0, while the other one moves towards the critical point J3 = 2.2. Fig. 7(b) shows the χJ3 results under different temperatures for J1 = 2.2. As the SF–D phase transition is known to be of the Berezinskii–Kosterlitz–Thouless (BKT) type [24,25], which cannot be detected by the fidelity susceptibility method [19,26], the χJ3 result does not give the D–SF phase transition information as well. There are two main drops in the χJ3 curve. As temperature is decreases, the one at around J3 = 1.15 keeps its position almost unchanged, while the other one moves towards the critical point J3 = 2.45 indicated in Fig. 3. The two drops indicate the phase transitions 1 and 2 of Fig. 2. Meanwhile, there appears another drop at around J3 = 1.5 with the decreasing of temperature. Because the phase transition 3 moves to the large J3 side as N increases as shown in Fig. 3(b), we think it is the precursor of the phase transition 3, and that is to say the threefold degenerate phase (IV) indeed exists. In addition, the product of magnetic susceptibility and temperature χh T at J1 = 0.5 are shown in Fig. 8, where J3 is set as 0.3, 0.9, and 1.5, respectively, which are correspondingly located at the regions (I), (III), and (II), respectively. The curves of J3 = 0.3 and J3 = 1.5 show a typical ferrimagnetic behavior—there is the upturn property leaving a minimum as temperature decreases; while the χh T curve at J3 = 0.9 shows no upturn behavior. These results further verify our conclusion that the regions (I) and (II) are ferrimagnetic states with the spin arrangement shown in Fig. 3.

Y.-C. Li / Physica A 389 (2010) 5550–5555

5555

4. Summary In summary, in terms of the SDGE method connecting with the analysis of energy-level properties for different phases, we study the quantum phase transitions of the VADC model. Due to spin frustration, there exist at least four different states: except for the ferrimagnetic, dimer, and spin–flute states, there is a threefold degenerate state (phase (IV)). When J1 is far away from J2 , the DM state is clearly separated from the ferri phase and the TD state is smoothly connected with the D state. We show the global phase diagram for the VADC model. In addition, we study the quantum phase transitions at finite temperatures for infinite systems by using the TMRG method. We calculate the J3 derivative of the driving term Hamiltonian χJ3 and magnetic susceptibility χh . The results verify the conclusions of the SDGEs. The χh results further determine the ferrimagnetic property of regions (I) and (II), and the χJ3 calculations show the existence of the threefold degenerate state (IV) as well. Acknowledgements This work was supported by the National Basic Research Program of China (973 Program) grant No. G2009CB929300 and the National Natural Science Foundation of China under Grant Nos. 60521001 and 60776061. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

[14] [15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25] [26]

S. Sachdev, Quantum Phase Transition, Cambridge University Press, Cambridge, UK, 1999. F. Plastina, T.J.G. Apollaro, Phys. Rev. Lett. 99 (2007) 177210. C. Cormick, J.P. Paz, 2007. e-print: arXiv:quant-ph/0709.2643. N.D. Mathur, F.M. Grosche, S.R. Julian, I.R. Walker, D.M. Freye, R.K.W. Haselwimmer, G.G. Lonzarich, Nature (London) 394 (1998) 39. K. Ueda, S. Miyahira, J. Phys.: Condens. Matter 11 (1999) L175. M. Hase, I. Terasaki, K. Uchinokura, Phys. Rev. Lett. 70 (1993) 3651. H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, H. Ohta, Phys. Rev. Lett. 94 (2005) 227201; H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, H. Ohta, Phys. Rev. Lett. 97 (2006) 089702. Y.C. Li, J. Appl. Phys. 102 (2007) 113907. K. Okamoto, T. Tonegawa, Y. Takahashi, M. Kaburagi, J. Phys.: Condens. Matter 11 (1999) 10485; K. Okamoto, T. Tonegawa, M. Kaburagi, J. Phys.: Condens. Matter 15 (2003) 5979. T. Tonegawa, K. Okamoto, T. Hikihara, Y. Takahashi, M. Kaburagi, J. Phys. Soc. Japan 69 (Suppl. A) (2000) 332; T. Tonegawa, K. Okamoto, T. Hikihara, Y. Takahashi, M. Kaburagi, J. Phys. Chem. Solids 62 (2001) 125. K. Sano, K. Takano, J. Phys. Soc. Japan 69 (2000) 2710. H.T. Wang, J. Phys.: Condens. Matter 14 (2002) 8033. P. Zanardi, N. Paunkovic, Phys. Rev. E 74 (2006) 031123; P. Zanardi, M. Cozzini, P. Giorda, J. Stat. Mech. 2 (2007) L02002; P. Zanardi, H.T. Quan, X.G. Wang, C.P. Sun, Phys. Rev. A 75 (2007) 032109. S.J. Gu, H.M. Kwok, W.Q. Ning, H.Q. Lin, Phys. Rev. B 77 (2008) 245109. M. Cozzini, P. Giorda, P. Zanardi, Phys. Rev. B 75 (2007) 014439; M. Cozzini, R. Ionicioiu, P. Zanardi, Phys. Rev. B 76 (2007) 104420. P. Buonsante, A. Vezzani, Phys. Rev. Lett. 98 (2007) 110601. W.L. You, Y.W. Li, S.J. Gu, Phys. Rev. E 76 (2007) 022101. S. Chen, L. Wang, S.J. Gu, Y. Wang, Phys. Rev. E 76 (2007) 061108. S. Chen, L. Wang, Y. Hao, Y. Wang, Phys. Rev. A 77 (2008) 032111. Y.C. Tzeng, M.F. Yang, Phys. Rev. A 77 (2008) 012311. K. Takano, K. Kubo, H. Sakamoto, J. Phys.: Condens. Matter 8 (1996) 6405. T. Xiang, X. Wang, in: I. Peschel, X. Wang, M. Kaulke, K. Hallberg (Eds.), Density-Matrix Renormalization: A New Numerical Method in Physics, Springer, New York, 1999, pp. 149–172; U. Schollwock, Rev. Modern Phys. 77 (2005) 259. Antôio S.F. Tenóio, R.R. Montenegro-Filho, M.D. Coutinho-Filho, Phys. Rev. B 80 (2009) 054409. V.L. Beresinskii, Sov. Phys. JETP 32 (1971) 493. J.M. Kosterlitz, D.J. Thouless, J. Phys. C 6 (1973) 1181; J.M. Kosterlitz, J. Phys. C 7 (1974) 1046. Y.C. Li, S.S. Li, Phys. Rev. B 78 (2008) 184412.