The effect of magnetic frustrations in a structure of Quasi-One-Dimensional Heisenberg antiferromagnetic chain

L. Cui a, F. Wang a, S.J. Zhang b, Y.J. Hu b,nQ1

a Suqian College, Fundamental Department, Suqian 223800, Chinab Hubei University of Automotive Technology, Shiyan 442002, China

a r t i c l e i n f o

Article history:Received 29 January 2014Received in revised form24 April 2014Accepted 28 April 2014

Keywords:Density-matrix renormalization groupmethodQuasi-One-Dimensional HeisenbergAntiferromagnetic chainSpin gap

a b s t r a c t

Using exact numerical diagonalization and density-matrix renormalization group method, we study theeffect of magnetic frustrations due to next-nearest-neighbor bonds in a structure of periodically dopingspins beside every spin side of the same sublattice of the 1D HAF linear chain, which is popularly knownas Quasi-One-Dimensional Heisenberg Antiferromagnetic chain. As a result of the frustrations, thequantum disordered phase (gapped) also appears in the quantum case, except that the ferrimagneticstate in the non-frustrations case and the caned phase appeared in the classical case. For quantumdisordered phase, tetramer–dimmer state is predominant and the spin gap is opened.

& 2014 Published by Elsevier B.V.

1. Introduction

Since the evidence of the existence of a short-range resonating-Valence-bond phase approach to high-temperature superconduc-tivity was found in a sufficiently frustrated Heisenberg model[1,2], a great deal of interest has been concentrated on thefrustrated spin system [3–5]. Meanwhile, compounds with veryinteresting frustrated magnetic lattices, like Rb2Cu2Mo3O12 [7],NaCu2O2 [8], and Cu3(CO3)2(OH)2 [9], have already been synthe-sized or found in the last decades.

Quasi-One-Dimensional (QOD) system are especially fascinat-ing theoretically because of their unexpected behavior whenviewed as interpolators between the one-dimensional(1D) anti-ferromagnetic Heisenberg chain and the 2D square analog. Whenthe magnetic frustrations are introduced in QOD HeisenbergAntiferromagnetic (HAF) chain, the situation is more complicateddue to the interplay of quantum fluctuations and frustrations.In general, frustration reduces the antiferromagnetic correlationsand in some cases may produce various exotic quantum groundstates such as the dimerized state [9,10], the spin-nematic state[11,12], or some sort of spin-liquid states [1,10]. Especially inter-esting are that frustrations may make spin system form a nearlyclosed state, or a cluster state, which has a simple product wavefunction. The dimerized state is the smallest cluster state, but

strong frustration can lead to larger local structures or clusters ofspins than a dimmer, such as tetramer (four-spin cluster state)–dimer state [13]. Although these cluster states correspond toparticular frustration parameters, they are good reference pointsin order to understand the global behavior of some frustratedHeisenberg models.

In our group's previous work [14], by using exact diagonaliza-tion, we have studied one spin-1/2 QOD HAF chain, i.e. periodicallydoping spins (side spin) beside every spin side of the samesublattice of the 1D HAF linear chain. And we have analyticallyproved that adding side spins can set up magnetic long-rangeorder, due to that the decay of spin–spin correlations withdistances slows down and becomes obviously flat in the range oflarge distances. In this paper, we introduce magnetic frustrationsdue to next-nearest-neighbor bonds in this spin-1/2 QOD HAFchain (see Fig. 1).

The system is described by the Hamiltonian

H¼ J1 ∑N

n ¼ 1ð S!An S

!Bnþ S

!Bn S!

Cnþ S!

Bn S!

Anþ1Þ

þ J2 ∑N

n ¼ 1ð S!An S

!Cnþ S

!Anþ1 S

!CnÞ; ð1Þ

where the integers n number N unit cells, S!

An; S!

Bn; S!

Cn are spinoperators on the sites of sublattice A, B, C in n-th unit cell. N isassumed to be even, so the total number of spins 3 N is a multipleof six. In the following, if not especially noted, periodic boundary

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Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/physb

Physica B

http://dx.doi.org/10.1016/j.physb.2014.04.0660921-4526/& 2014 Published by Elsevier B.V.

n Corresponding author. Tel.: þ86 05278420 0030.E-mail address: eric8222@126.com (Y.J. Hu).

Please cite this article as: L. Cui, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.04.066i

Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎

conditions are imposed. And we set J1¼1 and use the notationsα¼ J2/J1 for the frustration parameter.

There are two special cases of the present model, Eq. (1). First,when α¼0, we can readily show by the use of the Lieb-Mattistheorem [15] that the ground state is a ferrimagnetic (FRI) statewith the total spin ST¼N/2. We note that this value is one third ofthat of the maximum (ferromagnetic) value, 3 N/2. Second, thecase where α¼1 which also is a special case in the diamond-model, has already been investigated by Long et al. [16] and byTakano et al. [13]. They have shown that the exact solvable groundstate is the tetramer–dimer (TD) state with ST¼0. In the TD state(see Fig. 2), quadruplets SAn�1, SBn-1, SCn�1, and SAn (or SAn, SBn, SCn,and SAnþ1) of spins form singlet tetramers, and pairs SBn and SCn(SBnþ1 and SCnþ1) of spins form singlet dimmers. Here we focus onthe case of 0oαo1.

2. Ground state in the classical and quantum case

The classical phases of Heisenberg model may or may not havecounterparts in the quantum case, e.g., the square-lattice J1� J2mixed-spin Heisenberg model [17] exhibits similar classical mag-netic phases which persist in the quantum phase diagram, but thefrustrated mixed-spin ladder with diagonal exchange bonds [18]does not. What result of our discussed model, Eq. (1)?

In the classical case, all spin operators S!

An; S!

Bn; S!

Cn can beconsidered as classical vectors. We introduce two angles ðθ;φÞdescribing the classical phase. θðφÞ denotes the angle between thespins of SBn (SCn) and SAn. As a function of the frustration parameterα, the classical phase diagram of Eq. (1) exhibits the ferrimagneticstate (F), the canted state (C), and the collinear state (N) (see Fig. 3).The canted state phase is stable in the interval αc1oαoαc2.αc1 ¼ 1=3 and αc2 ¼ 1 are phase-transition points, separating thecanted state phase form the ferrimagnetic state phase ðθ;φÞ¼(π,0)and the collinear state phase ðθ;φÞ¼(π,π).

Nowwe turn to the quantum case. The ground state ST is a physicalquantity of fundamental importance since it is closely related to spinstatus (e.g. Ref. [19] has used it to find precise quantum-phasetransition critical points for the J1–J2 Heisenberg model with anothergeometric shape). So here we resort to exact diagonalization (ED)techniques to calculate the total spin of the ground state ST toinvestigate the quantum-phase transition of the present model.

Without frustration, the GS ST for our model takes N/2. Thenintroducing frustrations may alter the GS ST. Due to the principle ofclassical physics we can expect that the state with ST¼N/2 survivesup to some finite α [20]. While, frustration strong enough will

induce a disordered spin status. In this case, spins do not orientatein a certain polarization direction. Consequently, the GS ST willtake the he lowest possible value (e.g., as the total number of spinsis even, its value equals to zero).

According to physical consideration above and the correspond-ing classical phases, the quantum phases for our model areconnected with the following changes in the ground state ST:ST¼N/2 for 0rαoαc1 (Quantum Ferrimagnetic phase), 0oSToN/2 for αc1oαoαc2(Quantum Canted phase) and ST¼0 forαc2oαr1 (Quantum Disordered phase). The extrapolated datafor the total unit cells N¼4, 6, 8, 10 (i.e. the total number of spins is12, 18, 24, 30, respectively) give the results αc1¼0.4082 andαc2¼0.4110, showing that the region occupied by the quantum Cphase is absolutely narrowed but definitely finite.

Quantum ferrimagnetic phase and quantum canted phase arethe magnetic phases, which are ordered quantum phases. Followingthe terminology of Ref. [21], the quantum canted phase may becalled partially polarized ferromagnetic phase, as the GS ST is lessthan the value ST¼N/2 in the ferrimagnetic state phase. In Ref. [22],this quantum state may also be classified as a kind of ferromagneticLuttinger liquid. In this paper, our emphasis is on the region of thedisordered quantum phase.

3. Results from DMRG studies

In this section, we study the discussed model with openboundary conditions by employing infinite density-matrix renor-malization group (DMRG). The DMRG procedure follows the usualsteps for chains discussed in earlier papers [23,24], except that theQOD HAF chains with frustrations studied here are not symmetricbetween the left and right halves. Hence the density matrices forthese two halves have to be constructed at each iteration of thecalculations. In our calculation, the steps of iteration lets 100(i.e. 200 unit cells and 600 spin sites) and retained states m at eachDMRG iteration vary 80–100. We increase m to obtain moreaccurate results in the vicinity of critical points.

3.1. The GS energy per unit cell in the whole region ð0rαr1Þ

To detect quantum phase-transition critical point further, wecalculate the GS energy per unit cell (GSEPU) for the frustrationparameter αð0rαr1Þ. Though the GS ST changes from N/2 tozero in the whole region, the GS energy can be found in subspacewith Sz¼0. As shown in Fig. 4, GSERU curve has a maximum withα¼0.412. This is the order-disorder phase-transition point, whichis consistent with ED results αc2¼0.411.

3.2. The effect of frustrations for the magnetic phases

Here we introduce spin–spin correlations, which is defined asfollow:

Cαβðji� jjÞ ¼ ⟨Gj S!αi S!

βjjG⟩ ð2Þ

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AnS BnS

CnS

1+AnS1J

1J2J

Fig. 1. The spin-1/2 QOD HAF chain model studied in the paper. The circlerepresents a spin with magnitude S¼1/2, the solid line represents nearest-neighbor bond J1 and the dashed line represents next-nearest-neighbor bond J2(J14 J240). SAn, SBn, SCn are spins with magnitude S¼1/2 in the n-th unit cell whichconsisting of sublattice A, B, C.

AnS BnS

CnS

1+AnSAnS BnS

CnS

1+AnSAnSAnS BnSBnS

CnSCnS

1+AnS 1+AnSAnSAnS BnSBnS

CnSCnS

1+AnS 1+AnS

Fig. 2. The tetramer-dimer (TD) state in the special case where α¼1. The ovalrepresents a singlet dimer and the closed loop including four spins represents atetramer.

θ

φ

Fig. 3. Sketch of the classical ferrimagnetic (F), canted (C), and collinear (N) spinphases of the present model in one unit cell. θðφÞ denotes the angle between thespins of SBn (SCn) and SAn.

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where jG⟩ is the ground state of the system, i and j are the numberindex of the unit cell, α and β denote the sublattice A, B, C. Withthe parameter ji� jj increasing, the spin–spin correlationsCαβðji� jjÞ will converge to a constant value, namely Spin–spincorrelators Cαβ. We have calculated CAA, CBB and CAB, the spin–spincorrelators between pair of spins on the linear chain, to investigatethe effect of magnetic frustrations. In our DMRG procedure, wecompute these correlation functions from the unit cell inserted atthe last iteration when the system size reaches the thermody-namic limit scale, to minimize numerical errors.

The DMRG data for different spin–spin correlators (see Fig. 5)also give some additional evidence in favor of the suggested phasediagram by ED method in Section 2. The order–disorder phase-transition point αc2 also can be indicated as a point in Fig. 5 wherethe three correlators reduce abruptly to nearly zero. On the otherhand, we observe that the absolute value of three Spin–spincorrelators display only a slight monotonic decrease and remainfinite up to the transition point αc2, which indicating that themagnetic frustrations reduce the correlations between spins onthe linear chain and so reduce magnetic order.

3.3. The disordered quantum phase with ST¼0 ð0:412oαo1Þ

With period boundary conditions, the TD state of the specialcase where α¼1 for our model is two-fold degenerate. Whenemploying open boundary conditions, the TD state is nondegene-rate, i.e. the spins SA, SB, SC in odd unit cell and SA in even unit cellform singlet tetramers (therein, SB, SC in odd unit cell form atriplet), and SB,SC in even unit cell form singlet dimmers (seeFig. 2). The question is now, what will happen in this disorderregion where αa1?

To find out the GS spin structures where αa1, we define thefollowing three spin-state density projectors

(a) singlet projector in even unit cell

P1 ¼ jS2333 ¼ 0⟩⟨S

2333 ¼ 0j

where jS233 3 ¼ 0⟩

1ffiffiffi2

p ðjα23β

33 ⟩�jβ

23α

33 ⟩Þ

(b) triplet projector in odd unit cell

P2 ¼ jS23 ¼ 1⟩⟨S23 ¼ 1j where jS23 ¼ 1⟩

¼jα2α3⟩1ffiffi2

p ðjα2β3⟩þjβ2α3⟩Þjβ2β3⟩

8><>:

(c) tetramer–dimer (TD) projector in two unit cells

P3 ¼ jS1231

3 ¼ 0; S23 ¼ 1; S233 3 ¼ 0⟩⟨S

12313 ¼ 0; S23 ¼ 1; S

233 3 ¼ 0j

where

jS1231

3 ¼ 0; S23 ¼ 1; S233 3 ¼ 0⟩¼ jS

12313 ¼ 0; S23 ¼ 1⟩jS

233 3 ¼ 0⟩

¼ 1ffiffiffiffiffiffi24

p ðjα23β

33 ⟩�jβ

23α

33 ⟩Þnfjα1α2β3β1

3 ⟩þjβ1β2α3α13 ⟩þjα1β2α3β13 ⟩

þjβ1α2β3α13 ⟩�2jα1β2β3α1

3 ⟩�2jβ1α2α3β13 ⟩g

Note that, in the above expressions, subscripts 1, 2, 3 and 13;23;33

denote the spins of sublattice A, B, C in the odd and even unit cell,respectively.

In order to find out the statistical weight for a particular spinstate defined above, we can calculate its spin-state density definedbelow

⟨Pi⟩¼1N∑j⟨GjPiðjÞjG⟩; i¼ 1;2;3 ð3Þ

where jG⟩ is the ground state of the system, Pi is one of the threeprojector above, and j runs over all unit cell in the system. So ⟨P1⟩

represents the statistical weight of the spins SB and SC in the evenunit cell forming the singlet; ⟨P2⟩ represents the statistical weightof the spins SB and SC in the odd unit cell forming the triplet; and⟨P3⟩ represents the statistical weight of the six spins in two unitcells forming TD state.

Due to infinite DMRG procedure, we compute these three spin-state densities in the two unit cells inserted at the last iteration toreplace their expectation value for all unit cells in the system.Doing that, we can minimize numerical errors.

As shown in the Fig. 6, all three graphs show discontinuities atα¼ 0:411, the position at which the GSEPU reaches its maximum.On the left side of discontinuity, where the system is in themagnetic phases, ⟨P3⟩ equals to nearly zero; but On the right sideof discontinuity, where the system is in the disordered phase, thevalue of ⟨P3⟩ changes smoothly from about 0.77 to 1.00. So we canconclude that the spin structure, of which the six spins in everytwo unit cells forming TD state, is predominant for this disorderedregion.

Finally, we study the effect of the frustration on the spin gap fordisordered region. As result of the GS ST for the quantumdisordered phase taking the lowest possible value with ST¼0,the spin gap Δ is given by

Δ¼ E0ðSz ¼ 1Þ�E0ðSz ¼ 0Þwhere E0ðSz ¼ 1Þ and E0ðSz ¼ 0Þ represent the lowest energy inthe subspace Sz¼1 and Sz¼0, respectively. As shown in the Fig. 6,with α increasing from order–disorder phase-transition point

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Fig. 4. The ground state energy per unit cell (GSEPU) in subspace Sz¼0, as afunction of the frustration parameter αð0rαr1Þ.

Fig. 5. Spin–spin correlators CAA, CBB and CAB versus the frustration parameter α byinfinite DMRG method.

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αc2 ¼ 0:410, the spin gap gradually emerges. And remarkably thespin gap has maximum at α¼0.801. So we can conclude that thebreakdown of magnetic phase due to frustration is accompaniedby the opening of a spin gap.

4. Discussion and summary

In this article we have investigated the GS phase diagram of thepresent model (1) in the classical and quantum cases Fig. 7.

For one thing, we discuss the influence of quantum fluctuationson the GS properties of the present model. Our Heisenberg modelexhibits three classical phases, consisting of the ferrimagneticstate, the canted state, and the collinear state. In the quantumcase, the collinear state phase is narrowed within a very smallregion. And as a result of quantum fluctuations, the regionoriginally occupied by the classical collinear state phase is com-pletely replaced by quantum disordered phase. But the ferrimag-netic state phase survives quantum fluctuations and remains the

GS up to much stronger frustration than in the classical case. Thisphenomena again confirms the generally physical picture that forspin systems, quantum fluctuations favor spin structures withNeel state.

Secondly, we discuss the effect of frustrations due to next-nearest- neighbor bonds on our discussed model (1). In theclassical case, the canted spin phase appears as a result of themagnetic frustration. It must be noted that the C phase appearedhere is formed without explicit breaking of the SU(2) symmetry asopposed to in the bilayer quantum Heisenberg model subject to aperpendicular magnetic field [25]. In the quantum case, relativelymoderate frustrations are able to destroy the magnetic order andto stabilize the quantum disordered phase, the process of which isaccompanied by the opening of a spin gap. For the magneticphases, frustrations smoothly reduce the correlations betweenspins on the linear chain and so reduce magnetic order. For thedisordered phase, with the frustrations increased, the weight of TDspin-state density continuously increases from about 0.77 to 1.0.

Uncited Q2reference

[6].

Acknowledgment

This work is supported by the Ordinary University NationalScience Research Program of Jiangsu Province (12KJB140011), theScience and technology Program of Suqian and the ResearchFoundation of Suqian College.

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Fig. 6. Three spin-state densities ⟨P1⟩ ⟨P2⟩, ⟨P3⟩ versus the frustration parameter α inthe whole region ð0rαr1Þ.

Fig. 7. Spin gap Δ in the disordered region, as a function of the frustrationparameter α.

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