Quantum Phase Transitions of the Distorted Diamond Spin Chain
Tomosuke Zenda, Yuta Tachibana, Yuki Ueno, Kiyomi Okamoto, Tôru Sakai
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Quantum Phase Transitions of the Distorted Diamond Spin Chain
Tomosuke
Zenda , Yuta Tachibana , Yuki Ueno ,Kiyomi Okamoto and Tˆoru Sakai , Graduate School of Material Science,University of Hyogo, Hyogo 678-1297, Japan National Institutes for Quantum and Radiological Scienceand Technology (QST), SPring-8, Hyogo 679-5148, Japan (Dated: Received September 8, 2019)
Abstract
The frustrated quantum spin system on the distorted diamond chain lattice suitable for thealumoklyuchevskite is investigated using the numerical diagonalization of finite-size clusters andthe level spectroscopy analysis. It is found that this model exhibits three quantum phases; theferrimagnetic phase, the spin gap one, and the gapless Tomonaga-Luttinger liquid depending onthe exchange coupling parameters. The ground state phase diagram is presented. . INTRODUCTION Frustrated quantum spin systems have attracted a lot of interest in the field of stronglycorrelated electron systems. The S = 1 / (CO ) (OH) , called azurite . The previous theoretical work using the perturba-tion analysis, the numerical exact diagonalization of finite clusters, and the level spectroscopymethod, indicated that the system exhibits various quantum phases; the spin gap phase, theferromagnetic one, and gapless Tomonaga-Luttinger liquid (TLL) one in the ground state,depending on the exchange coupling parameters. Recently another candidate material ofthe distorted diamond spin chain was discovered. It is the compound K Cu AlO (SO ) ,called alumoklyuchevskite . This material has a different structure of the distortion fromazurite. Thus it would be useful to investigate the suitable theoretical model for alumok-lyuchevskite. In this paper, the S = 1 / II. MODEL
We investigate the model described by the Hamiltonian H = H + H (1) H = L X j { J S j, · S j, + J S j, · S j, + J S j, · S j, } (2) H = J L X j { S j, · S j +1 , + S j, · S j +1 , } (3)where S j,i is the spin-1/2 operator, J , J , and J are the coupling constants of the exchangeinteractions. The schematic picture of the model is shown in Fig. 1.For alumoklyuchevskite, it is thought that the interactions corresponding to four sides ofdiamond differ from one another. Since such a model, however, has many parameters, weuse a simplified model sketched in Fig.1. When J is much larger than other couplings, thespins coupled by J are going to form a singlet pairs, which make S j, spins nearly free. Ifthe direct or effective interactions between S j, spins are antiferromagnetic, the ground state2ill be the TLL state. This is the essential mechanism for the TLL ground state observed inalmoklyuchevskite. On the other hand, when J is much larger than other couplings, singletpairs locate at the J bonds, which yields nearly free S j, spins. If the direct or effectiveinteractions between S j, spins are antiferromagnetic, the ground state will be the TLL state,which is nothing but the essential mechanism for the TLL ground state of azurite. Thus ourmodel is a minimal model describing both TLL ground states of alumoklyuchevskite andazurite. We note that the direct interactions between nearly free spins are very importantto explain experimental results both of almoklyuchevskite and azurite . S j ,2 J J J S j ,1 S j ,3 FIG. 1: The model of the S = 1 / For L -unit systems, the lowest energy of H in the subspace where P j S zj = M , is denotedas E ( L, M ). The reduced magnetization m is defined as m = M/M s , where M s denotes thesaturation of the magnetization, namely M s = 2 L/ E ( L, M ) is calculatedby the Lanczos algorithm under the periodic boundary condition ( S L +1 ,i = S ,i ) for L =4,6 and 8. III. GROUND STATE PHASE DIAGRAM
We consider the ground state phase diagram of the model (1). Since the three differentexchange interactions J , J and J , we fix J = 1 and vary J and J in this paper. Onthe analogy of the azurite-type model, the ferrimagnetic, the spin gap and the gapless TLLphases are expected to appear. A. Ferrimagnetic phase
The ferrimagnetic phase is easily distinguished from other phases. In this phase thefinite magnetization m = 1 / J = 0 . J M = 0 and M = 4 for L = 8 are shown in Fig. 2.The phase boundary between the ferrimagnetic and singlet phases can be detected as theintersection of two energy levels. Since the phase boundary is almost independent of thesystem size, the phase boundary is estimated from the result for L = 8. J -14-12-10-8-6 E L =8 M =0 L =8 M =4 FIG. 2: J dependence of E ( L = 8 , M = 0) (black line) and E ( L = 8 , M = 4) (red line) for J = 0 .
5. The phase boundary between the ferrimagnetic and singlet phases can be detected asthe intersection of two energy levels.
B. Spin gap and TLL phases
In order to determine the phase boundary between the spin gap and the TLL phases,the level spectroscopy analysis is one of the best methods. According to this method, weshould compare the excitation energies of the lowest singlet excitation and the lowest tripletone. Namely, we define two excitation energies∆(
L, M = 0) = E ( L, M = 0) − E ( L, M = 0) , (4)∆( L, M = 1) = E ( L, M = 1) − E ( L, M = 0) , (5)where E ( L, M ) and E ( L, M ) are, respectively, the lowest energy and first excited energywithin the subspace of M for the L -unit system, The ground state is in the spin gap phase orthe TLL phase according as ∆( L, M = 0) > ∆( L, M = 1) or ∆(
L, M = 0) < ∆( L, M = 1).The J dependences of ∆’s with fixed J = 0 . L = 4, 6 and 8 are shown in Fig. 3.4ssuming the finite-size correction of the cross points between ∆( L, M = 0) and ∆(
L, M =1) is proportional to 1 /L , we estimate the phase boundary in the thermodynamic limit.This analysis indicates that the spin gap phase is adjacent to the ferrimagnetic phase. J ∆ L =4 M =0 L =4 M =1 L =6 M =0 L =6 M =1 L =8 M =0 L =8 M =1 FIG. 3: J dependences of ∆( L, M = 0) and ∆(
L, M = 1) with J = 0 . L = 4 (blacklines), 6 (blue lines) and 8 (red lines). Solid and dashed lines correspond to the ∆( L, M = 0) and∆(
L, M = 1), respectively.
C. Phase diagram
According to the above analyses, the ground state phase diagram is obtained as shown inFig. 4. As expected, it includes the ferrimagnetic, the spin gap and the TLL phases. Takano et al. indicated that the dimer-monomer state with high degeneracy is the exact groundstate on the line of J = 1 and J >
2. They also found that the doubly degenerate tetramer-dimer state is the ground state on the line of J = 1 and 0 . < J <
2. Reflecting this fact,our spin gap state is also doubly degenerate which is consistent with the level spectroscopymethod to determine the boundary between the spin-gap phase and the TLL phase.
IV. SUMMARY
Using the numerical exact diagonalization and the level spectroscopy analysis, the S = 1 / J J Ferri Spin Gap TLLTLL
Dimer-Monomer
Spin Gap
Tetramer-Dimer
FIG. 4: Ground state phase diagram of the present model. It includes the ferrimagnetic, the spingap and the TLL phases. On the line of J = 1, the dimer-monomer state is the exact ground statefor J > . < J < . TLL phases. We believe that the upper TLL state is attributed to nearly free S j, spins(alumoklyuchevskite type), while the lower TLL state to nearly free S j, spins (azurite type).More detailed analysis will be a future problem. Acknowledgment
This work was partly supported by JSPS KAKENHI, Grant Numbers 16K05419,16H01080 (J-Physics) and 18H04330 (J-Physics). A part of the computations was performedusing facilities of the Supercomputer Center, Institute for Solid State Physics, University ofTokyo, and the Computer Room, Yukawa Institute for Theoretical Physics, Kyoto Univer-sity. H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T.Juwai, and H. Ohta, Phys. Rev. Lett. , 227201 (2005). K. Okamoto T. Tonegawa, Y. Takahashi, and M. Kaburagi, J. Phys.: Condens. Matter , 10485(1999). M. Fujihala, H. Koorikawa, S. Mitsuda, M. Hagihala, H. Morodomi, T. Kawae, A. Mitsudo, andK. Kindo, J. Phys. Soc. Jpn. , 073702 (2015). K. Morita, M. Fujihala, H. Koorikawa, T. Sugimoto, S. Sota, S. Mitsuda and T. Tohyama, Phys.Rev. B , 184412 (2017). M. Fujihala, H. Koorikawa1, S. Mitsuda, K. Morita, T. Tohyama, K. Tomiyasu, A. Koda, H.Okabe, S. Itoh, T. Yokoo, S. Ibuka, M. Tadokoro, M. Itoh, H. Sagayama, R. Kumai and Y.Murakami, Sci. Rep. , 16785 (2017) H. Jeschke, I. Opahle, H. Kandpal, Roser Valent´ı, H. Das, T. Saha-Dasgupta, O. Janson, H.Rosner, A. Br¨uhl, B. Wolf, M.l Lang, J. Richter, S. Hu, X. Wang, R. Peters, T. Pruschke andA. Honecker, Phys. Rev. Lett. , 217201 (2011) K. Okamoto and K. Nomura, Phys. Lett. A , 433 (1992). K. Nomura and K. Okamoto, J. Phys. A: Math. Gen. , 5773 (1994). K. Kubo, K. Takano and H. Sakamoto, J. Phys: Cond. Matter , 6405 (1996), 6405 (1996)