Infinite Series of Ferrimagnetic Phases Emergent from the Gapless Spin Liquid Phase of Mixed Diamond Chains
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Journal of the Physical Society of Japan
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Infinite Series of Ferrimagnetic Phases Emergent from the Gapless SpinLiquid Phase of Mixed Diamond Chains
Kazuo Hida ∗ Professor Emeritus, Division of Material Science, Graduate School of Science and Engineering,Saitama University, Saitama, Saitama, 338-8570 (Received )
The ground-state phases of mixed diamond chains with (
S, τ (1) , τ (2) ) = (1 / , / , S is themagnitude of vertex spins, and τ (1) and τ (2) are those of apical spins, are investigated. The apical spins τ (1) and τ (2) are connected with each other by an exchange coupling λ . Other exchange couplings are setequal to unity. This model has an infinite number of local conservation laws. For large λ , the ground stateis equivalent to that of the uniform spin 1 / λ ≤
0, the ground state is a Lieb-Mattis ferrimagnetic phase with spontaneous magnetization m sp = 1per unit cell. For intermediate λ , we find a series of ferrimagnetic phases with m sp = 1 /p where p takespositive integer values. The phases with p ≥ p -fold translational symmetry. It is suggested that the phase with arbitrarily large p , namely infinitesimalspontaneous magnetization, is allowed as λ approaches the transition point to the gapless spin liquid phase.
1. Introduction
The quantum effects in low-dimensional frustratedmagnets have been extensively studied in recent con-densed matter physics.
1, 2)
Various exotic quantumphases emerge from the interplay of quantum fluctuationand frustration. Among them, the diamond chain, whose lattice structure is shown in Fig. 1, is known as amodel with an infinite number of local conservation laws.The ground states can be classified by the correspond-ing quantum numbers. If the two apical spins have equalmagnitudes, which is the case widely investigated, eachpair of apical spins can form a singlet dimer. It cuts thecorrelation between both sides and the ground state is adirect product of the cluster ground states separated bydimers.The ground states of spin-1/2 diamond chains havebeen investigated in Ref. 3. In addition to the spin clus-ter ground states, the ferrimagnetic state with sponta-neous magnetization m sp = 1 /
8, 9)
The ground states of spin-1 diamond chains have beenalso investigated in Refs. 3 and 4. In addition to the spincluster ground states, the nonmagnetic Haldane stateand the ferrimagnetic states with spontaneous magne-tization m sp = 1 and 1/2 are found. It should be notedthat the latter ferrimagnetic state is accompanied by aspontaneous translational symmetry breakdown. ∗ E-mail address: [email protected]
On the other hand, if the magnitudes of the two apicalspins are not equal to each other, they cannot form a sin-glet dimer. Hence, all spins in the chain inevitably form amany-body correlated state. In many cases, (quasi-)long-range order evolves including the vertex spins. As a sim-ple example of such cases, we investigate the mixed di-amond chain with apical spins of magnitude 1 and 1/2,and vertex spins, 1/2 in the present work. Remarkably,we find an infinite series of ferrimagnetic phases.This paper is organized as follows. In Sect. 2, the modelHamiltonian is presented. In Sect. 3, the ground-statephase diagram is determined numerically. The behaviorof the spontaneous magnetization in each phase is pre-sented and analyzed. The last section is devoted to asummary and discussion.
2. Hamiltonian
We consider the Hamiltonian H = L X l =1 h S l ( τ (1) l + τ (2) l )+ ( τ (1) l + τ (2) l ) S l +1 + λ τ (1) l τ (2) l i , (1)where S l , τ (1) l and τ (2) l are spin operators with magni-tudes S l = τ (1) l = 1 / τ (2) l = 1. The number of theunit cells is denoted by L , and the total number of sitesis 3 L . Here, the parameter λ controls the frustration asdepicted in Fig. 1.The Hamiltonian (1) has a series of local conservationlaws. To see it, we rewrite Eq. (1) in the form, H = L X l =1 (cid:20) S l T l + T l S l +1 + λ (cid:18) T l − (cid:19)(cid:21) , (2)
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FULL PAPERS S l λ τ l(1) S l+1 τ l(2)
11 S= τ (1) =1/2 τ (2) =1 Fig. 1.
Structure of the diamond chain investigated in this work S = τ (1) = 1 / τ (2) = 1. where the composite spin operators T l are defined as T l ≡ τ (1) l + τ (2) l ( l = 1 , , · · · , L ) . (3)Then, it is evident that[ T l , H ] = 0 ( l = 1 , , · · · , L ) . (4)Thus, we have L conserved quantities T l for all l . Bydefining the magnitude T l of the composite spin T l by T l = T l ( T l + 1), we have a set of good quantum numbers { T l ; l = 1 , , ...L } where T l = 1/2 and 3/2. The totalHilbert space of the Hamiltonian (2) consists of separatedsubspaces, each of which is specified by a definite set of { T l } , i.e., a sequence of 1/2 and 3/2. A pair of apical spinswith T l = 1 / T l = 3 /
3. Ground-State Phase Diagram λ ≫ λ ≫ ∀ l T l = 1 /
2. Hence, this model is equiva-lent to the spin-1/2 antiferromagnetic Heisenberg chainwhose ground state is a gapless spin liquid. λ ≪ λ ≪ ∀ l T l = 3 /
2. Hence, this model is equiv-alent to the spin-1/2-3/2 alternating antiferromagneticHeisenberg chain whose ground state is a ferrimagneticstate with spontaneous magnetization m sp = 1 per unitcell according to the Lieb-Mattis theorem. Here, m sp is defined by m sp = 1 L L X l =1 ( h S zl i + h T zl i ) (5)where hi denotes the expectation value in the groundstate. λ In the absence of spontaneous translational symmetrybreakdown, only above two phases are allowed. To pur-sue the possibility of other phases, we employ the finitesize DMRG method with the geometry of Fig. 2. Thecorresponding Hamiltonian is given by H = L X l =1 S l T l + L − X l =1 T l S l +1 + L X l =1 λ (cid:18) T l − (cid:19) . (6) S l λ τ l(1) S l+1 τ l(2) Fig. 2.
Lattice structure used for the finite size DMRG calcula-tion
This geometry is chosen to allow for the nonmagneticground state for λ ≫
1. Here and in what follows, thenumber of states χ kept in each subsystem in the DMRGcalculation ranged from 240 to 360. We find that theresults with χ = 240 are accurate enough in the presentwork.The ground-state energies of the Hamiltonian (6) upto L = 16 for all possible configurations { T l } are calcu-lated. The configurations that give the lowest energy areidentified for each λ . The spontaneous magnetization perunit cell is given by m sp = 1 L (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L X l =1 ( T l − S l ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (7)from the Lieb-Mattis theorem for the Hamiltonian (6).Figure 3 shows our numerical result for the λ -dependence of m sp . In addition to the two phases de-scribed in Sect. 3.1 and Sect 3.2, a quantized ferrimag-netic ground state with m sp = 1 / T l = q for odd l and T l = d for even l . We denote this configurationas (qd) L/ . In the thermodynamic limit, this configura-tion tends to the (qd) ∞ configuration. This configurationis equivalent to the (dq) ∞ configuration in the thermo-dynamic limit. Hence, this phase is doubly degenerateand is accompanied by the spontaneous breakdown oftwofold translational symmetry. In what follows, similarnotations are employed for other configurations. Withinthe finite-size DMRG calculation, the intermediate fer-rimagnetic phases are observed between this phase andthe nonmagnetic phase.For finite chains, however, the values of m sp are con-strained by the system size. Hence, it is not clear whether m sp is quantized or continuously varying with λ for0 < m sp < / { T l } for infinitechains. Hence, we analyze this regime in the followingway: We start with plausible candidates of periodic con-figurations of T l ’s and find the configuration that givesthe lowest energy ground state among them. This leadsto a stepwise λ -dependence of m sp . Then, we check thestability of these steps against the formation of defects.As plausible candidates of the ground states, we consider
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FULL PAPERS λ m sp L=8L=10L=12L=14L=14L=16 (qd) ∞ Fig. 3. λ -dependence of m sp calculated by the finite size DMRGmethod. S S S single q and (p−1) d’sp S’s (S=1/2)1 segment = p unit cells= qd p−1 q d d q
Fig. 4. (qd p − ) ∞ configuration. the configurations (qd p − ) ∞ with m sp = 1 /p that consistof an infinite array of segments qd p − with length of p unit cells as depicted in Fig. 4. These configurations canbe obtained from the configuration d ∞ corresponding tothe nonmagnetic ground state by inserting a q every p unit cells periodically. It should be remarked that thesestates are accompanied by the p -fold spontaneous trans-lational symmetry breakdown. Hereafter, this series ofconfigurations is called the main series configurations.The case p = 1 corresponds to the q ∞ configuration thatcorresponds to the Lieb-Mattis type ferrimagnetic phasewith m sp = 1.The λ -dependence of m sp in the main series is shownin Fig. 5. The spontaneous magnetization m sp rises con-tinuously from m sp = 0 at the critical value of λ givenby λ c ( ∞ ) ≡ lim p →∞ λ c ( p, p − ≃ .
807 (8)where the boundary between the (qd p − ) ∞ phase with m sp = 1 /p and (qd p − ) ∞ phase with m sp = 1 / ( p −
1) isdenoted by λ c ( p, p − p → ∞ iscarried out using the data for 38 ≥ p ≥
11 assuming the λ m sp p ≤ λ c ( ∞ )−~0.807p=2 p=1 p=3p=4 Fig. 5. λ -dependence of m sp for the main series configurationscalculated by the infinite size DMRG method. λ Fig. 6.
Extrapolation scheme of λ c ( p, p −
1) and λ c ( p + 1 , p ) to p → ∞ . The filled and open circles correspond to the fits by Eq.(9) and Eq. (10), respectively. The filled square is the extrapolatedvalue λ c ( ∞ ). following two asymptotic forms λ c ( p, p −
1) = λ c ( ∞ ) + C p + C p , (9) λ c ( p + 1 , p ) = λ c ( ∞ ) + C ′ p + C ′ p . (10)The extrapolation procedure is plotted in Fig. 6. Theboth extrapolations give the same value for λ c ( ∞ ) upto the above digit. It should be noted that Eq. (9) andEq. (10) correspond to the extrapolation of left and rightends of the steps, respectively. Using Eqs. (9) and (10),the λ -dependence of m sp is plotted down to m sp = 0 in
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FULL PAPERS λ m sp p ≤ λ c ( ∞ )−~0.807Eqs. (9),(10) Fig. 7. λ -dependence of m sp for the main series configurationsnear λ = λ c ( ∞ ). The open circles are the results of the infinitesize DMRG calculation. The small filled circles and solid lines areextrapolation by Eqs. (9) and (10). qd p−1 qd p−1 qd p−2 qd p−1 qd p−2 n segments=qd p−2 +(n−1) × qd p−1 (a) qd p−1 qd p−1 qd p qd p−1 qd p n segments=qd p +(n−1) × qd p−1 (b) Fig. 8. (a) (qd p − (qd p − ) n − ) ∞ configuration and (b)qd p ((qd p − ) n − ) ∞ configuration. Fig. 7.The remaining question is whether the intermediateconfigurations with 1 /p < m sp < / ( p −
1) can be aground state. To answer this question, it is necessary tocalculate the ground-state energies for all possible config-urations of { T l } , which is impossible. Hence, we confineourselves to the following configurations that are plausi-ble to compete with the (qd p − ) ∞ configurations.(1) (qd p − (qd p − ) n − ) ∞ configuration ( m sp = n/ ( p ( n −
1) + ( p − p − seg-ment is replaced by a qd p − segment per every n segments in the (qd p − ) ∞ configuration. For n = 1,this configuration reduces to the (qd p − ) ∞ configu-ration with m sp = 1 / ( p −
1) that corresponds to theneighboring step of the main series with higher m sp .For n = 2, this configuration reduces to that withalternating qd p − and qd p − segments. (2) (qd p (qd p − ) n − ) ∞ configuration ( m sp = n/ ( p ( n −
1) + ( p + 1))) depicted in Fig. 8(b):A qd p − segment is replaced by a qd p segment perevery n segments in the (qd p − ) ∞ configuration.For n = 1, this configuration reduces to the(qd p ) ∞ configuration with m sp = 1 / ( p + 1) thatcorresponds to the neighboring step of the mainseries with lower m sp . For n = 2, this configurationreduces to that with alternating qd p − and qd p segments.(3) The configurations with spatial periodicity less than12 and spontaneous magnetization m sp ≤ / p and n . We have numeri-cally confirmed that these states with 2 ≤ n ≤ ≤ p ≤
11. It is further confirmedthat those with 2 ≤ n ≤ ≤ p ≤
8. Within our numerical accuracy, we find noconfigurations that give lower energy than the main se-ries configurations. Within the available numerical data,the configurations with higher n are even less favorable.Hence, it is highly plausible that the state intermedi-ate m sp is not a ground state at least for p ≤
11. Also,configurations (3) do not give the ground state exceptfor the main series configurations. Thus, we expect thatonly the configurations in the main series are realized inthe ground state.
4. Summary and Discussion
The ground-state phases of diamond chains (1) with(
S, τ (1) , τ (2) ) = (1 / , / ,
1) are investigated. Betweenthe gapless spin liquid phase for large λ and Lieb-Mattisferrimagnetic phase with m sp = 1 for λ ≤
0, we find aseries of quantized ferrimagnetic phases with m sp = 1 /p where p takes all positive integer values.The λ -dependence of the spontaneous magnetization m sp is very different from other diamond chains with fer-rimagnetic ground states. Although the quantized ferri-magnetic phases are present even in undistorted diamondchains, the allowed values of spontaneous magnetizationare limited to several simple rational values.
3, 4)
There are some examples of ferrimagnetic groundstates of diamond chains that are induced by the lat-tice distortion.
5, 6)
In these cases, the ground state ofthe undistorted chain is a paramagnetic state consist-ing of clusters with finite magnetic moments. The sizesof the clusters are limited even in the absence of dis-tortion. The distortions induce ferromagnetic interac-tions between the cluster spins leading to the quantizedferrimagnetic phases. The quantum fluctuations of thelengths of clusters are also induced by distortion leadingto the partial ferrimagnetic phases. In the present case, the ferrimagnetic phases arepresent even in the absence of distortion. In contrast to
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FULL PAPERS the cases of Refs. 3–6, arbitrarily large segments are al-lowed leading to the infinitesimally small steps around λ = λ c ( ∞ ). However, no fluctuations of the lengths ofthe segments are allowed in the present model, since themagnitudes of the composite spins T l remain good quan-tum numbers. Hence, partial ferrimagnetic phases areabsent in contrast to the cases of Refs. 5 and 6.In the spin-1 alternating bond diamond chain withbond alternation δ , the nonmagnetic phase is equivalentto the ground state of the spin-1 alternating bond Heisen-berg chain with bond alternation δ . In this model, anintermediate ferrimagnetic phase is observed in the tinyregion close neighborhood of the point ( λ, δ ) = ( λ c , δ c ) ≃ (1 . , . In Ref. 7, it has beenspeculated that this region is the partial ferrimagneticphase. However, considering the similarity of the gaplessspin-liquid phase of the present model and the Haldane-dimer critical line of the spin-1 alternating bond diamondchain, it would be more reasonable to speculate that theinfinite series of quantized ferrimagnetic phases similarto those discussed in the present work is realized also inthis case. Unfortunately, the numerical confirmation isdifficult due to the smallness of the width of this region.As mentioned above, in some examples of the quan-tized ferrimagnetic phases in undistorted diamondchains, the allowed values of spontaneous magnetizationare limited to several rational values.
3, 4)
In these exam-ples, the nonmagnetic phases neighboring the ferrimag-netic phases are spin-gap phases. On the other hand, thenonmagnetic phase neighboring the ferrimagnetic phasein the present model with infinitesimal step is the gap-less spin liquid phase. This seems to suggest that theinfinitesimal energy scale of the gapless spin liquid phasehelps the emergence of the exotic ferrimagnetic phasewith infinitesimal spontaneous magnetization. A furtheranalytical approach would be required to get insight intothe physical implication of the present phenomenon.So far, the infinite series of ferrimagnetic phases pro-posed in this work have not been found in real mate-rials. However, since the gapless spin liquid phases are generic critical states in quantum spin chains, it wouldbe possible that these series of phases are realized inthe presence of appropriate frustrating exchange inter-actions. Nevertheless, in more realistic cases, the pertur-bation that does not preserve the conservation laws (4)is inevitable. In such cases, the infinitesimal structureof spontaneous magnetization might be smeared. In thiscontext, it would be an interesting problem to investi-gate the effect of lattice distortion in the present model.These studies are left for future investigation.A part of the numerical computation in this work hasbeen carried out using the facilities of the SupercomputerCenter, Institute for Solid State Physics, University ofTokyo, and Yukawa Institute Computer Facility at KyotoUniversity. Introduction to Frustrated Magnetism: Materials, Experi-ments, Theory , ed. C. Lacroix, P. Mendels, and F. Mila(Springer Series in Solid-State Sciences, Springer, Heidelberg,2011).2)
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