Frustrated magnetism of spin-1/2 Heisenberg diamond and octahedral chains as astatistical-mechanical monomer-dimer problem

Jozef Strecka,1, ∗ Taras Verkholyak,2 Johannes Richter,3, 4 Katarına Karl’ova,1 Oleg Derzhko,2 and Jurgen Schnack5

1Department of Theoretical Physics and Astrophysics, Faculty of Science,P. J. Safarik University, Park Angelinum 9, 04001 Kosice, Slovakia

2Institute for Condensed Matter Physics, NASU, Svientsitskii Street 1, 79011 L’viv, Ukraine3Institut fur Theoretische Physik, Otto-von-Guericke Universitat in Magdeburg, 39016 Magdeburg, Germany

4Max-Planck-Institut fur Physik Komplexer Systeme, 01187 Dresden, Germany5Department of Physics, Bielefeld University, P.O. Box 100131, D-33501 Bielefeld, Germany

(Dated: October 27, 2021)

It is evidenced that effective lattice-gas models of hard-core monomers and dimers afford a properdescription of low-temperature features of spin-1/2 Heisenberg diamond and octahedral chains.Besides monomeric particles assigned within the localized-magnon theory to bound one- and two-magnon eigenstates, the effective monomer-dimer lattice-gas model additionally includes dimericparticles assigned to a singlet-tetramer (singlet-hexamer) state as a cornerstone of dimer-tetramer(tetramer-hexamer) ground state of a spin-1/2 Heisenberg diamond (octahedral) chain. A feasibilityof the effective description is confirmed through the exact diagonalization and finite-temperatureLanczos methods. Both quantum spin chains display rich ground-state phase diagrams includingdiscontinuous as well as continuous field-driven phase transitions, whereby the specific heat showsin vicinity of the former phase transitions an extraordinary low-temperature peak coming from ahighly-degenerate manifold of low-lying excitations.

PACS numbers: 05.50.+q, 68.35.Rh, 75.10. Jm, 75.40.Cx, 75.50.NrKeywords: Heisenberg model, diamond chain, octahedral chain, monomer-dimer lattice gas, bound magnons

I. INTRODUCTION

Frustrated quantum Heisenberg spin systems remainfor a long time at the forefront of research interest, be-cause they often exhibit intriguing ordered or disorderedquantum ground states in addition to unconventionalmagnetic properties emergent at sufficiently low temper-atures [1, 2]. From this perspective, it is quite bizarrethat a more complex nature of exchange pathways initi-ating geometric spin frustration simultaneously offers agenuine possibility of finding quantum ground states offrustrated Heisenberg spin models by the use of rigorousmethods. Among a few highly celebrated examples offrustrated quantum Heisenberg spin models with exactlyknown ground states one could for instance mention thezig-zag ladder [3], the Shastry-Sutherland lattice [4], thedelta chain [5], and the one-dimensional chain of linkedtetrahedra [6].

Compared to this, it is generally much more diffi-cult to cope magnetic and thermodynamic properties offrustrated quantum Heisenberg spin models at low butnon-zero temperatures. The unbiased exact diagonaliza-tion (ED) [7, 8] or finite-temperature Lanczos method(FTLM) [9–14] are regrettably limited to relatively smallfinite-size Heisenberg spin systems involving at mosta few dozens of quantum spins. On the other hand,the large-scale simulations of frustrated Heisenberg spinmodels based on quantum Monte Carlo (QMC) methods

∗ jozef.strecka@upjs.sk

suffer from a serious negative-sign problem when per-formed in a standard basis, and one should accordinglyresort to more challenging sign-problem-free QMC com-putations performed in a dimer [15–19] or trimer [20]basis.

A flat band emergent in the one-magnon spectrum offrustrated Heisenberg spin systems bears evidence of amagnon, which is bound to a small portion of the frus-trated spin lattice due to a destructive quantum interfer-ence. Owing to this fact, the many-magnon eigenstates offrustrated Heisenberg spin models can be simply designedfrom a bound one-magnon eigenstate within the conceptof localized magnons establishing an effective lattice-gasmodel [21], which consists in an independent arrange-ment of bound magnons on trapping cells of frustratedspin lattices [22–24].

It should be pointed out, however, that the effectivelattice-gas models developed within the standard formu-lation of the localized-magnon theory are usually eligiblefor a description of frustrated quantum Heisenberg spinsystems only in a high-field region sufficiently close tothe saturation field [22–24]. One may fortunately avoidthis shortcoming in a special subclass of the frustratedquantum Heisenberg spin models satisfying local spin-conservation laws. It has been actually verified thatan additional counting of the lowest-energy bound two-magnon eigenstate paves the way towards a complete de-scription of low-temperature magnetization curves andthermodynamics of highly frustrated Heisenberg octahe-dral chains from zero up to the saturation field [25–28].

In the present paper we will generalize the localized-magnon theories established previously for the spin-

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1/2 Heisenberg diamond and octahedral chains by ex-tending their validity to a less frustrated parame-ter space including dimer-tetramer [29] and tetramer-hexamer [25] ground states, respectively. It is worth-while to remark that the spin-1/2 Heisenberg diamondchain bears a close relation to several copper-basedmagnetic compounds such as azurite Cu3(CO3)2(OH)2

[30, 31], A3Cu3AlO2(SO4)4 (A = K, Rb, and Cs) [32–34], [Cu3(OH)2(CH3COO)(H2O)4](RSO3)2 (RSO3 = or-ganic sulfonate anions) [35] and Cu3 (CH3COO)4(OH)2

· 5H2O [36]. Although we do not have at present anyknowledge about a possible experimental realization ofthe spin-1/2 Heisenberg octahedral chain a relatively nu-merous family of hexanuclear complexes with the mag-netic core of a discrete copper-based octahedron {Cu6}[37–40] could be used for a brick-and-mortar synthesisof a one-dimensional copper polymeric chain of corner-sharing octahedra. A closely related extended magneticstructure with the architecture of a three-dimensionalnetwork of corner-sharing octahedra {Cu6} is for instancerealized in the copper-based metal-organic framework[Cu3(tpt)4](ClO4)3 [41]. Last but not least, an artifi-cial design of the spin-1/2 Heisenberg octahedral chainachieved through magnetic atom-trap lattices is also fea-sible with regard to a recent successful realization of sev-eral related one-dimensional magnetic structures such asladders and diamond spin chains [42].

The outline of this paper is as follows. The over-all ground-state phase diagrams of the spin-1/2 Heisen-berg diamond and octahedral chains will be presentedin Sec. II and IV together with a few basic steps elu-cidating its construction from exact and density-matrixrenormalization group (DMRG) calculations of the effec-tive mixed-spin Heisenberg chains. The spin-1/2 Heisen-berg diamond and octahedral chains will be reformulatedwithin the modified localized-magnon theory establishingtheir connection to a one-dimensional lattice-gas model ofhard-core monomers and dimers in Sec. III and V, wherean eligibility of the effective description will be also cor-roborated through a detailed comparison with numericaldata obtained from the full ED and FTLM. The paperends up with a brief summary of the most importantfindings in Sec. VI, where a few future outlooks are alsopresented.

II. GROUND STATES OF A SPIN-1/2HEISENBERG DIAMOND CHAIN

First, let us consider the spin-1/2 Heisenberg diamondchain, which is schematically illustrated in Fig. 1 andmathematically defined through the Hamiltonian

H =

N∑j=1

[J1(S1,j + S1,j+1)·(S2,j + S3,j)

+J2S2,j ·S3,j − h3∑i=1

Szi,j

], (1)

J1

J2

S S1, 1, +1j j

S2,j

S3,j

J1

J1J1

FIG. 1. A schematic illustration of the spin-1/2 Heisenbergdiamond chain including the notation for the lattice sites andassumed coupling constants.

where Si,j (i = 1, 2, 3; j = 1, . . . , N) denotes spin-1/2 op-erator assigned to a lattice site unambiguously given bytwo subscripts, the first subscript specifies spins from agiven unit cell and the second subscript determines theunit cell itself. The coupling constant J1 > 0 standsfor the antiferromagnetic interaction between nearest-neighbor monomeric (S1,j) and dimeric (S2,j , S3,j) spins,while the coupling constant J2 > 0 accounts for the an-tiferromagnetic intradimer interaction being responsiblefor a geometric spin frustration. The Zeeman term h ≥ 0accounts for an effect of the external magnetic field andfinally, the periodic boundary condition is imposed forsimplicity S1,N ≡ S1,1.

The ground-state phase diagram of the spin-1/2Heisenberg diamond chain given by Hamiltonian (1) canbe obtained by combining three complementary tech-niques. One may adapt variational arguments (see Ap-pendix in Ref. [43]) in order to rigorously prove in thehighly frustrated parameter region J2 > 2J1 and lowenough magnetic fields h < J1 + J2 the monomer-dimer(MD) ground state schematically illustrated in Fig. 2and mathematically given by the eigenvector

|MD〉=

N∏j=1

|S1,j〉⊗1√2

(|↑2,j↓3,j〉−|↓2,j↑3,j〉), h = 0,

N∏j=1

|↑1,j〉⊗1√2

(|↑2,j↓3,j〉−|↓2,j↑3,j〉), h > 0,

(2)

where |S1,j〉 denotes any out of two available states |↑1,j〉and |↓1,j〉 of the monomeric spins. It is worthwhile toremark that the MD ground state (2) is macroscopicallydegenerate at zero magnetic field due to frustrated (para-magnetic) character of all monomeric spins S1,j , whichare consequently perfectly aligned into the magnetic-fielddirection once the external field is turned on. Anotherindependent confirmation of the MD ground state (2) isprovided by the localized-magnon approach, which re-stricts its presence in the highly frustrated parameterspace J2 > 2J1 to sufficiently low magnetic fields h < hsnot exceeding the saturation field hs = J1 +J2 associatedwith a field-driven phase transition to the fully saturatedferromagnetic phase [23, 44, 45]. It could be thus con-cluded that the variational and localized-magnon theoriesgive evidence of the MD ground state (2) from zero upto the saturation field in the highly frustrated parameterregion J2 > 2J1.

In the rest of the parameter space J2 < 2J1, the groundstates of the spin-1/2 Heisenberg diamond chain cannotbe simply found on the grounds of exact analytical cal-

3

FIG. 2. A schematic illustration of the monomer-dimerground state (2), which is responsible for occurrence of in-termediate one-third plateau in a zero-temperature magneti-zation curve. An oval denotes a singlet-dimer state.

composite

S S1, 1, +1j c,j jS

FIG. 3. A schematic illustration of the effective mixed spin-(1/2, Sc,j) Heisenberg chains, in which the composite spinsmay acquire the values Sc,j = 0 or 1 for a diamond chain andSc,j = 0, 1 or 2 for an octahedral chain.

culations. To this end, it is convenient to rewrite thezero-field part of the Hamiltonian (1) into the followingequivalent form

H = J1

N∑j=1

(S1,j+S1,j+1)·Sc,j +J2

2

N∑j=1

(S2c,j−

3

2

),(3)

which directly implies a local conservation of the compos-ite spin Sc,j = S2,j + S3,j on vertical dimers of a spin-1/2Heisenberg diamond chain. The composite spin on ver-tical dimers is accordingly conserved quantity with welldefined quantum spin numbers Sc,j = 0 or 1. Hence, itfollows that all ground states of the spin-1/2 Heisenbergdiamond chain can be derived from the lowest-energyeigenstates of effective mixed spin-(1/2, Sc,j) Heisenbergchains schematically illustrated in Fig. 3 when perform-ing appropriate shift of their eigenenergies according tothe second term of the Hamiltonian (3). Note further-more that the z-component of the total spin operator

SzT =∑Nj=1

∑3i=1 S

zi,j commutes with the Hamiltonian

(1), which means that most preferable energy eigenvaluesof the spin-1/2 Heisenberg diamond chain in a nonzeromagnetic field directly follow from their zero-field coun-terparts when performing just a trivial shift by the re-spective Zeeman’s term

E(SzT , h 6= 0) = E(SzT , h = 0)− hSzT . (4)

Although one should generally find out lowest-energyeigenstates of effective mixed spin-(1/2, Sc,j) Heisenbergchains (3) with all possible combinations of the compos-ite spins (∀j Sc,j = 0 or 1) it is often sufficient to con-sider just a few particular cases with quantum spin num-bers that do not break translational symmetry or at mostlimit the spontaneous breaking of translational symme-try to a few unit cells. By inspection we have foundthat all ground states of the spin-1/2 Heisenberg dia-mond chain stem from the effective mixed spin-(1/2, 0)or mixed spin-(1/2, 1) Heisenberg chains without trans-lationally broken symmetry or the effective mixed spin-(1/2, 1, 1/2, 0) Heisenberg chain with a period doubling.The lowest-energy eigenstate of the effective mixed spin-(1/2, 0) Heisenberg chain with the unique choice of zero

composite spin ∀j Sc,j = 0 naturally corresponds to theMD ground state (2) with the exact energy

E1/2−0(N,SzT = N/2) = −3

4NJ2 −

1

2Nh, (5)

because the singlet-dimer state at vertical bonds assignedto zero composite spin decouples all correlations betweenthe dimeric and monomeric spins. It is worthwhile to re-mark, moreover, that the effective mixed-spin Heisenbergchains with regularly alternating zero composite spins arealways fragmented into smaller quantum spin clusters, forwhich the lowest-energy eigenstates can be rather easilycalculated by the ED method. On the other hand, thefragmentation does not occur for the other unique choiceof the composite spins ∀j Sc,j = 1, which contrarily givesrise to highly correlated collective eigenstates with theenergy eigenvalues governed by the formula

E1/2−1(N,SzT )=NJ1ε1/2−1(N,SzT )+1

4NJ2−hSzT ,(6)

where ε1/2−1(N,SzT ) denotes the energy per unit cell ofthe effective mixed spin-(1/2, 1) Heisenberg chain withunit coupling constant, the number of unit cells N and z-component of the total spin SzT . The mixed spin-(1/2, 1)Heisenberg chain exhibits at low enough magnetic fieldsthe gapped ferrimagnetic phase with a z-component ofthe total spin SzT = (1 − 1

2 ) ×N = N/2 due to orderingof energy levels imposed by the Lieb-Mattis theorem [46],whereby the eigenstates with higher values of the totalspin momentum SzT > N/2 form a continuous band per-tinent to the gapless Tomonaga-Luttinger quantum spinliquid emergent at higher magnetic fields when an energygap above the Lieb-Mattis ferrimagnetic phase closes [47–51]. To gain the respective eigenenergies ε1/2−1(N,SzT )of the effective mixed spin-(1/2, 1) Heisenberg chain onehas to resort to state-of-the-art numerical calculations.To this end, we have implemented DMRG simulations ofthe effective mixed spin-(1/2, 1) Heisenberg chain withup to 120 spins (N = 60 unit cells, which correspond tothe diamond chain of 180 spins) within the open-sourcesoftware Algorithms and Libraries for Physics Simulation(ALPS) project [53]. Finally, another available groundstate relates to the lowest-energy eigenstate of the effec-tive mixed spin-(1/2, 1, 1/2, 0) Heisenberg chain, wheresinglet and polarized triplet states regularly alternate onvertical dimers. The singlet-dimer states break the ef-fective mixed spin-(1/2, 1, 1/2, 0) Heisenberg chain intononinteracting three-spin clusters, which also turn out tobe in the singlet state within the lowest-energy eigenstatewith the energy

E1/2−1−1/2−0(N,SzT = 0) = −1

4NJ2 −NJ1. (7)

This lowest-energy eigenstate thus corresponds to the sin-glet tetramer-dimer (TD) phase of the spin-1/2 Heisen-berg diamond chain, which is schematically depicted inFig. 4 and mathematically given by the eigenvector

4

|TD〉 =

N/2∏j=1

1√3

[| ↑1,2j−1↓2,2j−1↑3,2j−1↓1,2j〉+ |↓1,2j−1↑2,2j−1↓3,2j−1↑1,2j〉 −

1

2

(|↑1,2j−1↑2,2j−1↓3,2j−1↓1,2j〉

+ |↑1,2j−1↓2,2j−1↓3,2j−1↑1,2j〉+ |↓1,2j−1↑2,2j−1↑3,2j−1↓1,2j〉+ |↓1,2j−1↓2,2j−1↑3,2j−1↑1,2j〉)]

⊗ 1√2

(|↑2,2j↓3,2j〉 − |↓2,2j↑3,2j〉

). (8)

FIG. 4. A schematic illustration of the singlet tetramer-dimerphase given by the eigenvector (8). Large and small ovalsrepresent regularly alternating singlet-tetramer and singlet-dimer states.

Note that the singlet TD phase is two-fold degener-ate, because another linearly independent eigenstate withthe same energy can be obtained from the eigenvec-tor (8) by interchanging singlet-tetramer and singlet-dimer states on odd and even unit cells. The TD phaseemerges just if the interaction ratio is from the interval0.91 < J2/J1 < 2.0 and the magnetic field is sufficientlysmall h/J1 < 0.55, whereby the highest possible fieldvalue for existence of the TH ground state is achieved forJ2/J1 ≈ 1.45, see Fig. 5.

The ground-state phase diagram constructed from allaforedescribed lowest-energy eigenstates of the spin-1/2Heisenberg diamond chain is depicted in Fig. 5 in the in-teraction ratio versus magnetic field plane. In total, thedisplayed phase diagram involves five different groundstates: the gapped Lieb-Mattis ferrimagnetic phase, thegapless Tomonaga-Luttinger quantum spin liquid, thefully polarized ferromagnetic phase, the MD phase andthe TD phase. Two horizontal phase boundaries de-limiting the Tomonaga-Luttinger quantum spin liquidmark continuous field-driven quantum phase transitions,while all other phase boundaries denote discontinuousfield-driven phase transitions accompanied with discon-tinuous magnetization jumps. The magnetization corre-sponding to the ferrimagnetic phase and the MD phaseequals to one-third of the saturation magnetization, whilethe TD phase has zero magnetization owing to the sin-glet character of regularly alternating singlet tetramersand singlet dimers. Contrary to this, the magnetizationof the Tomonaga-Luttinger quantum spin liquid contin-uously rises upon increasing of the magnetic field dueto the gapless character of this quantum ground state.To the best of our knowledge, the overall ground-statephase diagram of the symmetric spin-1/2 Heisenberg di-amond chain has not been reported in the literature yet,while the zero-field phase boundaries between the ferri-magnetic, TD and MD phases are in a perfect agreementwith the former results reported by Takano, Kubo andSakamoto [29].

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

tetramer-dimer

saturation (1/2-1)sat

spin liquid (1/2-1)

monomer-dimer 1/3-plateau (1/2-0)sat

ferrimagnetic 1/3-plateau (1/2-1)

h / J

1J2 / J1

FIG. 5. The ground-state phase diagram of the spin-1/2Heisenberg diamond chain in J2/J1 − h/J plane. The num-bers in parentheses determine spin values within the effectivemixed spin-(1/2, Sc) Heisenberg chain.

III. HEISENBERG DIAMOND CHAIN ASMONOMER-DIMER PROBLEM

In this section, we will turn our attention to a descrip-tion of low-temperature magnetization curves and ther-modynamics of the spin-1/2 Heisenberg diamond chain.It is noteworthy that the standard localized-magnontheory based on a classical lattice-gas model of hard-core monomers offers a satisfactory description of low-temperature magneto-thermodynamics of the spin-1/2Heisenberg diamond chain just in the highly frustratedparameter region J2/J1 > 2 [23, 44, 45], whereas our aimis to proceed beyond this simple theory. The low-energyfeatures of the spin-1/2 Heisenberg diamond chain willbe effectively described by the one-dimensional lattice-gas model of hard-core monomers and dimers, which willallow us to extend the range of validity of the localized-magnon approach to a less frustrated parameter spaceinvolving the singlet TD phase (8). The monomer-dimerlattice-gas model defined on an auxiliary one-dimensionallattice includes hard-core monomeric particles assignedto the singlet-dimer states on vertical bonds

|m〉j =1√2

(|↑2,j↓3,j〉 − |↓2,j↑3,j〉), (9)

as well as, hard-core dimeric particles assigned to thesinglet-tetramer states of diamond (four-spin) clusters

5

monomer dimer

dimer tetramersinglet

FIG. 6. One typical permissible state of the spin-1/2 Heisen-berg diamond chain and its equivalent representation withintwo-component lattice-gas model of hard-core monomers anddimers. Small red spheres of the auxiliary lattice correspondto the monomeric spins S1,j , while large blue spheres of theauxiliary lattice correspond to the composite spins Sc,j as-signed to the vertical dimers S2,j − S3,j .

being a cornerstone of the TD ground state (8)

|d〉j =1√3

(|↑1,j↓2,j↑3,j↓1,j+1〉+ |↓1,j↑2,j↓3,j↑1,j+1〉

)− 1√

12

(|↑1,j↑2,j↓3,j↓1,j+1〉+ |↑1,j↓2,j↓3,j↑1,j+1〉

+|↓1,j↑2,j↑3,j↓1,j+1〉+ |↓1,j↓2,j↑3,j↑1,j+1〉).

(10)

One typical permissible state of the spin-1/2 Heisen-berg diamond chain and its equivalent representationwithin the developed monomer-dimer lattice-gas modelis displayed in Fig. 6. Small red circles of the auxil-iary lattice correspond to lattice sites of the monomericspins S1,j and large blue circles of the auxiliary latticecorrespond to lattice positions of the composite spinsSc,j = S2,j + S3,j of the vertical dimers. The compositespins corresponding to the vertical dimers are accessi-ble to the fully polarized ferromagnetic state acting as areference vacuum state, the singlet-dimer state (9) repre-sented by a monomeric particle schematically shown bya large green circle and the singlet-tetramer state (10)represented by a dimeric particle (large violet rectangle)incorporating two enclosing monomeric spins S1,j andS1,j+1 as well. If the energy is defined relative with re-spect to the fully polarized ferromagnetic state servingas a reference state, then, one should assign the chemicalpotential µ1 = J1+J2−h to the monomeric particles con-nected with the singlet-dimer state (9) and the chemicalpotential µ2 = 3J1 − h to the dimeric particles associ-ated with the singlet-tetramer state (10). The effectiveHamiltonian of the one-dimensional lattice-gas model ofhard-core monomers and dimers consequently reads

Heff = E0FM − h

N∑j=1

[1 + Sz1,j(1− dj−1)(1− dj)]

− µ1

N∑j=1

mj − µ2

N∑j=1

dj , (11)

where E0FM = NJ1 + NJ2/4 refers to the zero-field en-

ergy of the fully polarized ferromagnetic state, the sec-

ond term is the respective Zeeman’s energy; mj = 0, 1and dj = 0, 1 are occupation numbers of the hard-coremonomeric and dimeric particles, respectively. The par-tition function of the monomer-dimer lattice-gas modeldefined through the Hamiltonian (11) can be written inthe compact form

Z = e−βE0FM

∑{Sz

1,j}

∑{mj}

∑{dj}

N∏j=1

1

4dj(1−dj−1dj)(1−mjdj)

× exp{β[µ1mj+µ2dj+hSz1,j(1−dj−1)(1−dj)]}. (12)

Here, β = 1/(kBT ), kB is the Boltzmann constant, Tis absolute temperature, the summation

∑{Sz

1,j}runs

over spin states of all monomeric spins S1,j , the sum-mations

∑{mj} and

∑{dj} are carried out over all pos-

sible values of the respective occupation numbers. Thefactor (1 − dj−1dj)(1 − mjdj) ensures a hard-core con-straint forbidding overlap and/or double occupancy ofauxiliary lattice sites by the monomeric and dimeric par-ticles, while the factor 1

4djis the correction term for the

singlet-tetramer state (10) that would be without thisfactor accounted four times when one performs the sum-mation over spin states of two monomeric spins S1,j andS1,j+1. The important feature of the partition function(12) is the possibility to sum over all the states of themonomeric spins S1,j and hard-core monomer particlesmj independently. In fact, the complete hard-core condi-tions between the quasi-particles should also include theprojection operator [1− (1−mi)di+1][1− di(1−mi+1)],which ensures that the singlet-plaquette state (di parti-cle) cannot be followed by the fully polarized state on theadjacent dimer. The energy of the latter state is muchlarger that the energy scale of allowed states present inthe effective Hamiltonian (11) and for simplicity, it canbe completely left out from consideration. After perform-ing the summations

∑{Sz

1,j}and

∑{mj} the expression

behind the product symbol can be expressed solely interms of the occupation numbers of the dimeric particles

T(dj , dj+1)=1

4dj(1−dj−1dj)2eβh+βµ2dj [1+(1−dj)eβµ1 ]

× cosh

[βh

2(1−dj−1)(1−dj)

]. (13)

The expression (13) can be in turn identified as the trans-fer matrix, which allows a simple calculation of the parti-tion function (12) within the transfer-matrix method [52]exploiting a consecutive summation over the occupationnumbers of the dimeric particles

Z = e−βE0FM

∑{dj}

N∏j=1

T(dj−1, dj) = e−βE0FM Tr TN

= e−βE0FM(λN+ + λN− ). (14)

The final expression for the partition function (14)is given through two eigenvalues λ± obtained after a

6

straightforward diagonalization of two-by-two transfermatrix (13)

λ± = eβh

{cosh

(βh

2

)Ξd ±

√[cosh

(βh

2

)Ξd

]2

+ eβµ2Ξd

},

Ξd = 1 + eβµ1 . (15)

The free-energy density of the finite-size spin-1/2 Heisen-berg diamond chain normalized per spin consequentlyreads

f3N = −kBT1

3NlnZ

=1

3

(J1 +

J2

4

)− 1

3NkBT ln(λN+ + λN− ). (16)

The simpler expression can be acquired for the free-energy density of the infinite spin-1/2 Heisenberg dia-mond chain, which depends in the thermodynamic limitN → ∞ solely on the larger eigenvalue of the transfer-matrix

f∞ = −kBT limN→∞

1

3NlnZ =

1

3

(J1+

J2

4

)−1

3kBT lnλ+.

(17)

It is noteworthy that the final formulas (16) and (17)for the free-energy density allow a straightforward calcu-lation of the magnetization, susceptibility, entropy andspecific heat for a finite and infinite spin-1/2 Heisenbergdiamond chain, respectively. To verify a reliability of theeffective description based on the monomer-dimer lattice-gas model (11) we will compare as-obtained magnetiza-tion and specific-heat data with the extensive numericalcalculations employing the ED and FTLM implementedwithin the open-source ALPS [53] and Spinpack [54, 55]softwares.

First, let us comment the most interesting results forthe highly frustrated spin-1/2 Heisenberg diamond chainwith J2/J1 > 2. For illustrative purposes, magnetizationcurves of the spin-1/2 Heisenberg diamond chain are dis-played in the left panel of Fig. 7 for the fixed valueof the interaction ratio J2/J1 = 2.1 and four differenttemperatures. The magnetization data derived from thefinite-size formula (16) for the free-energy density of theeffective monomer-dimer model (11) are compared in thisfigure with the full ED data for 18 spins (N = 6 unit cells,upper panel) and FTLM results for 30 spins (N = 10 unitcells, lower panel) of the spin-1/2 Heisenberg diamondchain (1). As one can see, the magnetization curves ac-quired from the monomer-dimer model (11) perfectly co-incide at sufficiently low temperatures kBT/J1 . 0.2 withaccurate numerical results for the magnetization curvesof the spin-1/2 Heisenberg diamond chain. Generally,the intermediate 1/3-plateau resembling the MD groundstate gradually shrinks upon increasing of temperatureand the magnetization curve becomes smoother. To gaina more complete understanding, the magnetic-field vari-ations of the specific heat of the spin-1/2 Heisenberg di-amond chain are depicted in the right panel of Fig. 7 for

the same fixed value of the interaction ratio J2/J1 = 2.1and three different temperatures. It is obvious from thisfigure that the perfect agreement between the specific-heat data stemming from the effective monomer-dimermodel (11) and the precise numerical data acquired forthe spin-1/2 Heisenberg diamond chain (1) is limited tomuch lower temperatures kBT/J1 . 0.05, while the dis-crepancy becomes more pronounced at higher tempera-tures even if the effective description still at least quali-tatively reproduces the most essential features of the ac-curate numerical data. It should be also pointed outthat the magnetization and specific heat of the spin-1/2Heisenberg diamond chain do not show at low enoughtemperatures almost any finite-size dependence (confrontdata for the diamond chain with 18 and 30 spins), be-cause most thermally populated excited states are fromthe monomeric universality class.

Magnetization process and specific heat are displayedin Fig. 8 for the spin-1/2 Heiseberg diamond chain with arelative strength of the interaction constants J2/J1 = 1.8promoting a moderately strong spin frustration. Underthis condition, the spin-1/2 Heisenberg diamond chainexhibits in a low-field region zero and one-third mag-netization plateaus, which can be ascribed to the TDand MD ground states, respectively. The magnetiza-tion curves extracted from the effective lattice-gas modelof hard-core monomers and dimers (11) are in excellentagreement with the precise numerical data obtained fromthe full ED (upper panel) or FTLM (lower panel) up tomoderate temperatures kBT/J1 . 0.2, see the left panelin Fig. 8. The magnetic-field dependences of the spe-cific heat of the spin-1/2 Heisenberg diamond chain areplotted on the right panel of Fig. 8 for the same valueof the interaction ratio J2/J1 = 1.8 and three differenttemperatures. It is evident that the specific heat dis-plays in a low-field region a pronounced double-peak de-pendence, which appears in a vicinity of the field-drivenphase transition between the TD and MD ground statesemergent at h/J1 = 0.2. Unlike the previous case, thepeak heights of the specific heat above and below therelevant field-induced phase transition are not the samedue to difference in the relative degeneracy of the TD andMD ground states. It turns out that the low-temperaturedependences of the specific heat stemming from the ef-fective monomer-dimer model (11) satisfactorily repro-duce the accurate numerical data gained for the spin-1/2Heisenberg diamond chain (1) with the help of ED andFTLM only at low enough temperatures kBT/J1 . 0.05,whereas there is only qualitative rather than quantitativeagreement at higher temperatures kBT/J1 & 0.1. An-other interesting observation is that the less frustratedspin-1/2 Heisenberg diamond chain with J2/J1 = 1.8shows a marked finite-size effect of the specific heat whenboth peak heights are at sufficiently low temperaturessomewhat higher for the diamond chain with 30 spinsthan for 18 spins.

7

0.0 0.5 1.02.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0 ED full (18 spins) MDM (18 spins)

0.1

0.01

kBT / J1 = 0.2

m /

ms

h / J1

J2 / J1 = 2.1

0.05

0.0 0.4 0.8 2.5 3.0 3.5 4.00.00

0.05

0.10

0.15

0.20 ED-full (18 spins) MDM (18 spins)

0.050.01

kBT / J1 = 0.1

c / k

B

h / J1

J2 / J1 = 2.1

0.0 0.5 1.02.0 2.5 3.0 3.5 4.00.0

0.2

0.4

0.6

0.8

1.0

FTLM (30 spins) MDM (30 spins)

0.1

0.01

kBT / J1 = 0.2

m /

ms

h / J1

J2 / J1 = 2.1 0.05

0.0 0.4 0.8 2.5 3.0 3.5 4.00.00

0.05

0.10

0.15

0.20 FTLM (30 spins) MDM (30 spins)

0.050.01

kBT / J1 = 0.1

c / k

B

h / J1

J2 / J1 = 2.1

FIG. 7. (Left panel) Magnetization curves of the spin-1/2 Heisenberg diamond chain with J2/J1 = 2.1 obtained from theeffective monomer-dimer model (MDM) versus the ED data for 18 spins (upper panel) and the FTLM data for 30 spins (lowerpanel) of the original model (1). (Right panel) The same as in the left panel but for the specific heat. Thin black lines showanalytical results derived from the effective monomer-dimer model (11), while colored symbols refer to the ED (upper panel)or FTLM (lower panel). Note that there is an axis break in the middle of intermediate 1/3-plateau.

IV. GROUND STATES OF A SPIN-1/2HEISENBERG OCTAHEDRAL CHAIN

Another paradigmatic example to be consideredhereafter is the spin-1/2 Heisenberg octahedral chainschematically shown in Fig. 9 and defined via the Hamil-tonian

H =

N∑j=1

[J1(S1,j + S1,j+1)·(S2,j + S3,j + S4,j + S5,j)

+ J2(S2,j ·S3,j + S3,j ·S4,j + S4,j ·S5,j + S5,j ·S2,j)

− h5∑i=1

Szi,j

]. (18)

Here, the coupling constant J2 denotes antiferromagneticexchange interaction between nearest-neighbor spinsfrom square plaquettes and the coupling constant J1

accounts for the nearest-neighbor interaction betweenmonomeric spins and square-plaquette spins. The pa-rameter h stands for the external magnetic field, N is

the total number of unit cells and the periodic boundarycondition is assumed S1,N ≡ S1,1 for simplicity. A fullground-state phase diagram of the spin-1/2 Heisenbergoctahedral chain (18) was established using the varia-tional technique, localized-magnon approach and DMRGcalculations in our previous paper [25], to which readersinterested in further calculation details are referred to.To make our subsequent discussion self-contained we willmerely quote only the main outcomes of this calculationprocedure, which was accomplished in a similar fashionas thoroughly described for the spin-1/2 Heisenberg dia-mond chain in the preceding part.

The variational method provides for the spin-1/2Heisenberg octahedral chain (18) an exact evidence ofthe monomer-tetramer (MT) ground state in the highlyfrustrated parameter region J2 > 2J1 and low enoughmagnetic fields h < J1 + J2. The MT ground state is forbetter illustration schematically drawn in Fig. 10 andcan be defined through the eigenvector with character ofa tensor-product state of uncorrelated monomeric spinsand singlet tetramers on square plaquettes

8

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

ED-full (18 spins) MDM (18 spins)

0.2

0.1

kBT / J1 = 0.01

m /

ms

h / J1

J2 / J1 = 1.8

0.05

0.0 0.2 0.4 0.60.00

0.05

0.10

0.15

0.20 ED-full (18 spins) MDM (18 spins)

0.05

0.01

kB T /J1=0.1

c / k

B

h / J1

J2 / J1 = 1.8

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.1

0.2

0.3

0.4

FTLM (30 spins) MDM (30 spins)

0.2

0.1

kBT / J1 = 0.01

m /

ms

h / J1

J2 / J1 = 1.8

0.05

0.0 0.2 0.4 0.60.00

0.05

0.10

0.15

0.20 FTLM (30 spins) MDM (30 spins)

0.05

0.01

kB T /J1=0.1

c / k

B

h / J1

J2 / J1 = 1.8

FIG. 8. (Left panel) Magnetization curves of the spin-1/2 Heisenberg diamond chain with J2/J1 = 1.8 obtained from theeffective monomer-dimer model (MDM) versus the ED data for 18 spins (upper panel) and the FTLM data for 30 spins (lowerpanel) of the original model (1). (Right panel) The same as in the left panel but for the specific heat. Thin black lines showanalytical results derived from the effective monomer-dimer model (11), while colored symbols refer to the ED (upper panel)or FTLM (lower panel).

J1

J2

S S1, 1, +1j j

S2,j

S3,j

S4,j

S5,j

FIG. 9. A schematic illustration of the spin-1/2 Heisenbergoctahedral chain including notation for the lattice sites andassumed coupling constants.

|MT〉 =

N∏j=1

|S1,j〉⊗1√3

[1

2

(|↑2,j↑3,j↓4,j↓5,j〉+ |↑2,j↓3,j↓4,j↑5,j〉+ |↓2,j↑3,j↑4,j↓5,j〉+ |↓2,j↓3,j↑4,j↑5,j〉

)−|↑2,j↓3,j↑4,j↓5,j〉 − |↓2,j↑3,j↓4,j↑5,j〉

], h = 0,

N∏j=1

|↑1,j〉⊗1√3

[1

2

(|↑2,j↑3,j↓4,j↓5,j〉+ |↑2,j↓3,j↓4,j↑5,j〉+ |↓2,j↑3,j↑4,j↓5,j〉+ |↓2,j↓3,j↑4,j↑5,j〉

)−|↑2,j↓3,j↑4,j↓5,j〉 − |↓2,j↑3,j↓4,j↑5,j〉

], h > 0,

(19)

where |S1,j〉 denotes any out of two available states |↑1,j〉 and |↓1,j〉 of the monomeric spins. It is noteworthy that

9

FIG. 10. A schematic illustration of the monomer-tetramerground state (19), which is responsible for occurrence of theintermediate one-fifth plateau in a zero-temperature magne-tization curve. An oval denotes a singlet-tetramer state.

+

++

+

+

+

+

+

_

_

_

_

_

_

_

_

FIG. 11. A schematic illustration of the magnon-crystalground state (20), which is responsible for occurrence of theintermediate three-fifths plateau in a zero-temperature mag-netization curve. An oval denotes a magnon bound to a squareplaquette due to a destructive quantum interference caused byalternating signs of the probability amplitudes.

the monomeric spins are within the MT phase (19) com-pletely free to flip at zero magnetic field due to a spinfrustration invoked by a singlet-tetramer state, which canbe alternatively viewed as a bound two-magnon eigen-state of a square plaquette. Owing to this fact, themonomeric spins become fully polarized by any nonzeroexternal magnetic field and the MT ground state (19) willconsequently manifest itself in a zero-temperature mag-netization curve as the intermediate one-fifth plateau.

On the other hand, the localized-magnon theory de-veloped for the spin-1/2 Heisenberg octahedral chainfurnishes a rigorous proof for the magnon-crystal (MC)ground state, which is schematically shown in Fig. 11and mathematically given by the eigenvector

|MC〉=N∏j=1

|↑1,j〉⊗1

2

(|↓2,j↑3,j↑4,j↑5,j〉−|↑2,j↓3,j↑4,j↑5,j〉

+|↑2,j↑3,j↓4,j↑5,j〉−|↑2,j↑3,j↑4,j↓5,j〉).

(20)

The MC phase (20) has the character of a tensor-product state of the fully polarized monomeric spins andthe bound one-magnon eigenstates, which trap a singlemagnon within each square plaquette due to a destruc-tive quantum interference ensured by alternating signs

of the probability amplitudes. Note furthermore thatthe MC ground state emerges in the highly frustratedparameter region J2 > 2J1 and sufficiently high mag-netic fields that however do not exceed the saturationfield h < hs = J1 + 2J2.

In the rest of the parameter space J2 < 2J1 the groundstates of the spin-1/2 Heisenberg octahedral chain can befound by rewriting the zero-field part of the Hamiltonian(18) using the composite spin operator of the square pla-

quette Sc,j = S2,j + S3,j + S4,j + S5,j and two compos-ite operators for spin pairs from opposite corners of thesquare plaquette S24,j = S2,j+S4,j and S35,j = S3,j+S5,j

H = J1

N∑j=1

(S1,j + S1,j+1)·Sc,j

+J2

2

N∑j=1

(S2c,j − S2

24,j − S235,j). (21)

The composite spin operators Sc,j , S24,j and S35,j com-mute with the Hamiltonian (21) and thus, they cor-respond to conserved quantities with well defined spinquantum numbers Sc,j = 0, 1 or 2; S24,j = 0 or 1;S35,j = 0 or 1. The spin-1/2 Heisenberg octahedral chainbecomes equivalent to the effective mixed spin-(1/2, Sc)Heisenberg chains (see Fig. 3) whose eigenenergies arejust trivially shifted due to the second term of the Hamil-tonian (21) depending on the quantum spin numbers Sc,j ,S24,j and S35,j .

The lowest-energy eigenstates of the effective mixedspin-(1/2, Sc) Heisenberg chains were previously ob-tained by extensive DMRG simulations [25] implementedwithin the open-source ALPS software [53]. If the fourspins forming a square plaquette are in the singlet-tetramer state, i.e. the composite spin of a square pla-quette becomes zero Sc,j = 0, then, the spin-1/2 Heisen-berg octahedral chain is broken into smaller fragmentsdue to a lack of spin-spin correlations across this squareplaquette. A few exact fragmented ground states canbe found using this procedure. In this way one recoversfor instance the MT ground state (19) by considering ∀jSc,j = 0 and the MC ground state (20) when assuming∀j Sc,j = 1. In addition, the regular alternation of thecomposite spin values Sc,2j−1 = 1 and Sc,2j = 0 withinthe effective mixed spin-(1/2,1,1/2,0) Heisenberg chainresults in the singlet tetramer-hexamer (TH) phase givenby the eigenvector

|TH〉 =

N/2∏j=1

1√3

[|↑2,2j−1↓3,2j−1↑4,2j−1↓5,2j−1〉+ |↓2,2j−1↑3,2j−1↓4,2j−1↑5,2j−1〉 −

1

2|↑2,2j−1↑3,2j−1↓4,2j−1↓5,2j−1〉

−1

2

(|↑2,2j−1↓3,2j−1↓4,2j−1↑5,2j−1〉+ |↓2,2j−1↑3,2j−1↑4,2j−1↓5,2j−1〉+ |↓2,2j−1↓3,2j−1↑4,2j−1↑5,2j−1〉

)]⊗ 1√

12

(|↑1,2j↑2,2j↓3,2j↑4,2j↓5,2j↓1,2j+1〉+ |↓1,2j↑2,2j↓3,2j↑4,2j↓5,2j↑1,2j+1〉+ |↑1,2j↓2,2j↑3,2j↓4,2j↓5,2j↑1,2j+1〉

+|↑1,2j↓2,2j↓3,2j↓4,2j↑5,2j↑1,2j+1〉+ |↓1,2j↑2,2j↑3,2j↓4,2j↑5,2j↓1,2j+1〉+ |↓1,2j↓2,2j↑3,2j↑4,2j↑5,2j↓1,2j+1〉

10

−|↑1,2j↓2,2j↑3,2j↓4,2j↑5,2j↓1,2j+1〉 − |↓1,2j↓2,2j↑3,2j↓4,2j↑5,2j↑1,2j+1〉 − |↑1,2j↑2,2j↓3,2j↓4,2j↓5,2j↑1,2j+1〉

−|↑1,2j↓2,2j↓3,2j↑4,2j↓5,2j↑1,2j+1〉 − |↓1,2j↑2,2j↑3,2j↑4,2j↓5,2j↓1,2j+1〉 − |↓1,2j↑2,2j↓3,2j↑4,2j↑5,2j↓1,2j+1〉).

(22)

FIG. 12. A schematic illustration of the singlet tetramer-hexamer phase (22), which leads to occurrence of zero plateauin a zero-temperature magnetization curve. Small and largeovals correspond to the singlet-tetramer and singlet-hexamerstate, respectively.

0.0 0.5 1.0 1.5 2.0 2.5 3.00

1

2

3

4

5

6

2/5-plateau

ferrimagnetic 1/5-plateau (1/2-1)

monomer-tetramer 1/5-plateau (1/2-0)sat

ferrimagnetic 3/5-plateau (1/2-2)

spin liquid (1/2-1)

spin liquid (1/2-2)

saturation (1/2-2)sat

bound magnon crystal

3/5-plateau

(1

/2-1) sat

h / J

1

J2 / J1

tetramer-hexamer

FIG. 13. The ground-state phase diagram of the spin-1/2Heisenberg octahedral chain in J2/J1 − h/J1 plane.

The singlet TH phase is in fact two-fold degenerate, be-cause another linearly independent eigenstate with thesame energy can be obtained from the eigenvector (22)by interchanging singlet-tetramer and singlet-hexamerstates on odd and even unit cells. The TH phaseemerges just if the interaction ratio is from the interval0.91 < J2/J1 < 2.0 and the magnetic field is sufficientlysmall h/J1 < 0.55, whereby the highest possible fieldvalue for existence of the TH ground state is achieved forJ2/J1 ≈ 1.45, see Fig. 13.

As can be seen in Fig. 13, the overall ground-state phase diagram of the spin-1/2 Heisenberg octahe-dral chain reveals a great diversity of remarkable quan-tum ground states including two ferrimagnetic Lieb-Mattis phases originating from the mixed spin-(1/2, 1)and mixed spin-(1/2, 2) Heisenberg chains, respectively,two quantum spin-liquid phases derived from the samecouple of the mixed-spin Heisenberg chains closing anenergy gap above the Lieb-Mattis ferrimagnetic phases,the gapful ferrimagnetic phase with a translationally bro-ken symmetry, the fully polarized ferromagnetic phaseand three additional fragmented ground states denoted

monomer dimer monomer(2) (1)

tetramer hexamer magnonsinglet bound

FIG. 14. A schematic illustration of one paradigmatic eigen-state of the spin-1/2 Heisenberg octahedral chain and itsequivalent representation within the three-component lattice-gas model of hard-core monomers and dimers. Small redspheres of the auxiliary lattice correspond to the monomericspins S1,j , while large blue spheres of the auxiliary latticecorrespond to the composite spins Sc,j assigned to the squareplaquettes occupied by one of two monomeric particles or adimeric particle.

as the MT phase [see Eq. (19) and Fig. 10], the boundMC phase [see Eq. (20) and Fig. 11)] and the singlet THphase [see Eq. (22) and Fig. 12].

V. HEISENBERG OCTAHEDRAL CHAIN ASMONOMER-DIMER PROBLEM

In the following part we will investigate in detail low-temperature magnetization curves and thermodynam-ics of the spin-1/2 Heisenberg octahedral chain as ex-tracted from the anticipated one-dimensional lattice-gasmodel of hard-core monomers and dimers. Recently,we have convincingly evidenced that the simpler ver-sion of the localized-magnon theory based on an effectiveone-dimensional lattice gas including two types of hard-core monomeric particles ascribed to bound one- andtwo-magnon eigenstates of square plaquettes provideswithin the highly frustrated parameter region J2/J1 > 2a proper description of low-temperature magnetizationcurves and thermodynamics of the spin-1/2 Heisenbergoctahedral chain from zero up to the saturation field[25, 26]. The main goal of the present work is to ex-tend the validity of the localized-magnon theory to a lessfrustrated parameter space additionally involving the sin-glet TH phase (22). The idea and calculation procedureis quite similar as previously elucidated for the spin-1/2Heisenberg diamond chain.

The low-energy physics of the spin-1/2 Heisenberg oc-tahedral chain will be effectively described by the lattice-gas model of hard-core monomers and dimers defined onauxiliary one-dimensional lattice, which may include themonomeric particle assigned to the singlet-tetramer (i.e.

11

bound two-magnon) eigenstate of a square plaquette

|m〉j =1√3

[|↑2,j↓3,j↑4,j↓5,j〉+ |↓2,j↑3,j↓4,j↑5,j〉

−1

2

(|↑2,j↑3,j↓4,j↓5,j〉+ |↑2,j↓3,j↓4,j↑5,j〉

)(23)

−1

2

(|↓2,j↑3,j↑4,j↓5,j〉+ |↓2,j↓3,j↑4,j↑5,j〉

)],

the monomeric particle assigned to the bound one-magnon eigenstate of a square plaquette

|n〉j =1

2

(|↓2,j↑3,j↑4,j↑5,j〉 − |↑2,j↓3,j↑4,j↑5,j〉

+|↑2,j↑3,j↓4,j↑5,j〉 − |↑2,j↑3,j↑4,j↓5,j〉), (24)

as well as, the dimeric particle assigned to the singlet-hexamer state of a octahedron (six-spin) cluster being acornerstone of the TH ground states (22)

|d〉j =1√12

(|↑1,2j↑2,2j↓3,2j↑4,2j↓5,2j↓1,2j+1〉

+|↓1,2j↑2,2j↓3,2j↑4,2j↓5,2j↑1,2j+1〉+|↑1,2j↓2,2j↑3,2j↓4,2j↓5,2j↑1,2j+1〉+|↑1,2j↓2,2j↓3,2j↓4,2j↑5,2j↑1,2j+1〉+|↓1,2j↑2,2j↑3,2j↓4,2j↑5,2j↓1,2j+1〉+|↓1,2j↓2,2j↑3,2j↑4,2j↑5,2j↓1,2j+1〉−|↑1,2j↓2,2j↑3,2j↓4,2j↑5,2j↓1,2j+1〉−|↓1,2j↓2,2j↑3,2j↓4,2j↑5,2j↑1,2j+1〉−|↑1,2j↑2,2j↓3,2j↓4,2j↓5,2j↑1,2j+1〉−|↑1,2j↓2,2j↓3,2j↑4,2j↓5,2j↑1,2j+1〉−|↓1,2j↑2,2j↑3,2j↑4,2j↓5,2j↓1,2j+1〉

−|↓1,2j↑2,2j↓3,2j↑4,2j↑5,2j↓1,2j+1〉). (25)

One paradigmatic example of the feasible eigenstate ofthe spin-1/2 Heisenberg octahedral chain and its equiva-lent representation within the designed monomer-dimerlattice-gas model is illustrated in Fig. 14. Small redspheres of the auxiliary lattice correspond to lattice sitesof the monomeric spins S1,j and large blue spheres ofthe auxiliary lattice correspond to lattice positions ofthe composite spins Sc,j = S2,j + S3,j + S4,j + S5,j ofthe square plaquettes, which are available to the fullypolarized ferromagnetic state acting as a reference vac-uum state, the singlet-tetramer state (23) represented bythe monomeric particle shown as a large green sphere,the bound one-magnon state (24) represented by themonomeric particle depicted as a large brown sphere,and the singlet-hexamer state (25) represented by thedimeric particle displayed as a large violet rectangle in-volving two neighboring monomeric spins S1,j and S1,j+1

as well. In what follows, the energy of the square pla-quettes is determined relative with respect to the fullypolarized ferromagnetic state serving as a reference stateand consequently, one should assign the chemical poten-

tial µ(2)1 = 2J1 + 3J2 − 2h to the monomeric particles

assigned to the bound two-magnon state (23), the chem-

ical potential µ(1)1 = J1 + 2J2 − h to the monomeric par-

ticles connected to the bound one-magnon state (24) andthe chemical potential µ2 = 3J1 − h to the dimeric par-ticles ascribed to the singlet-hexamer state (25). Theeffective monomer-dimer lattice-gas model can be thendefined through the Hamiltonian

H = E0FM − h

N∑j=1

[2 + Sz1,j(1− dj−1)(1− dj)]

−µ(2)1

N∑j=1

mj − µ(1)1

N∑j=1

nj − µ2

N∑j=1

dj , (26)

where E0FM =2NJ1+NJ2 is zero-field energy of the fully

polarized ferromagnetic state, the second term is the Zee-man’s term, mj = 0, 1, nj = 0, 1 and dj = 0, 1 label occu-pation numbers for three kinds of particles additionallyobeying hard-core constraint since each square plaque-tte of the spin-1/2 Heisenberg octahedral chain may beat most in one out of three considered eigenstates (23)-(25). The partition function of the effective monomer-dimer lattice-gas model given by the Hamiltonian (26)can be calculated from the formula

Z=e−βE0FM

∑{Sz

1,j}

∑{mj}

∑{nj}

∑{dj}

N∏j=1

1

4dj(1−dj−1dj)(1−mjdj)

×(1−njdj)(1−mjnj) exp[β(µ(2)1 mj + µ

(1)1 nj + µ2dj)]

×exp[βhSz1,j(1− dj−1)(1− dj)], (27)

which already includes the factor (1 − dj−1dj)(1 −mjdj)(1−njdj)(1−mjnj) ensuring a hard-core rule thatprevents overlap and/or double occupancy of auxiliarylattice sites by the monomeric and dimeric particles.The other factor 1

4djentering into the partition func-

tion (27) is the correction term for the singlet-hexamerstate (25), which avoids fourfold counting of this eigen-state when performing the summation over spin statesof two monomeric spins S1,j and S1,j+1. As in Sec. III,we note that the partition function above contains theforbidden configurations when the singlet-plaquette stateis followed by the fully polarized or bound one-magnonstate on the neighboring dimer. But since the energy ofthese two configurations are rather high, they have noeffect on the low-temperature thermodynamic propertiesstudied here. After performing the summations

∑{Sz

1,j},∑

{mj} and∑{nj} over sets of the monomeric spins and

monomeric hard-core particles one gets behind the prod-uct symbol the expression depending exclusively on theoccupation numbers of the dimeric particles

T(dj , dj+1)=1

4dj(1−dj−1dj) cosh

[βh

2(1−dj−1)(1−dj)

]× 2e2βh+βµ2dj [1 + (1− dj)(eβµ

(1)1 + eβµ

(2)1 )],

(28)

12

which can be repeatedly identified with the transfer ma-trix [cf. (13)] further simplifying the exact calculation ofthe partition function according to the standard proce-dure

Z = e−βE0FM

∑{dj}

N∏j=1

T(dj−1, dj) = e−βE0FM Tr TN

= e−βE0FM(λN+ + λN− ). (29)

The final formula for the partition function (29) is ex-pressed in terms of eigenvalues λ± of two-by-two transfermatrix (28) easily accessible through a direct diagonal-ization

λ± = e2βh

{cosh

(βh

2

)Ξo ±

√[cosh

(βh

2

)Ξo

]2

+ eβµ2 Ξo

},

Ξo = 1 + eβµ(1)1 + eβµ

(2)1 , (30)

which have quite analogous form to the one previouslyreported for the spin-1/2 Heisenberg diamond chain, cf.(15). The free-energy density of the finite-size spin-1/2Heisenberg octahedral chain normalized per spin takesthe following form

f5N = −kBT1

5NlnZ

=1

5(2J1 + J2)− 1

5NkBT ln(λN+ + λN− ), (31)

which can be further simplified in the thermodynamiclimit N →∞ when the free-energy density of the infinitespin-1/2 Heisenberg octahedral chain per spin is solelyexpressed in terms of the larger transfer-matrix eigen-value

f∞ = −kBT limN→∞

1

5NlnZ =

1

5(2J1+J2)− 1

5kBT lnλ+.

(32)

The final formulas (31) and (32) for the free-energy den-sity may be subsequently employed for a straightforwardcalculation of the magnetization, susceptibility, entropyand specific heat of a finite and infinite spin-1/2 Heisen-berg octahedral chain, respectively. The correctness ofthe effective description based on the monomer-dimerlattice-gas model (26) will be exemplified through a com-parison of the as-obtained magnetization and specific-heat data with extensive numerical calculations employ-ing the full ED and FTLM implemented within the open-source ALPS [53] and Spinpack [54, 55] softwares.

First, the magnetization and specific heat of the spin-1/2 Heisenberg octahedral chain are plotted in Fig. 15as a function of the magnetic field for the interactionratio J2/J1 = 2.1. The spin-1/2 Heisenberg octahedralchain apparently exhibits two intermediate magnetiza-tion plateaus at one-fifth and three-fifths of the satura-tion magnetization, which can be attributed to the MTand MC ground states unambiguously given by the eigen-vectors (19) and (20), respectively. It directly follows

from Fig. 15 that the magnetization curves descendedfrom the effective monomer-dimer lattice-gas model arein a perfect agreement with the full ED data for 20 spinsand FTLM data for 30 spins up to moderate temperaturekBT/J1 ≈ 0.5. The perfect concordance is a direct con-sequence of a proper counting of localized many-magnoneigenstates (including among others two realized MT andMC ground states), which form a low-lying part of the en-ergy spectrum being most relevant at low enough temper-atures. On the other hand, the magnetic-field variationsof the specific heat display more pronounced discrepan-cies already at much lower temperatures kBT/J1 & 0.1,above which collective quantum states not accounted inthe anticipated monomer-dimer lattice-gas model are suf-ficiently thermally populated in order to cause conspic-uous uprise with respect to the specific-heat value origi-nating entirely from the many-magnon eigenstates. Inspite of this shortcoming, the monomer-dimer lattice-gas model at least qualitatively reproduces a double-peakfeature of the specific heat observable in a vicinity of allfield-driven phase transitions at low and moderate tem-peratures.

Last but not least, the low-field part of magnetizationcurves of the spin-1/2 Heisenberg octahedral chain is de-picted in the left panel of Fig. 16 for the less frustratedparameter region J2/J1 = 1.8, where the investigatedquantum spin chain undergoes a field-driven phase tran-sition from the TH state (22) to the MT state (19). It canbe understood from Fig. 16 that the effective monomer-dimer model (26) not only properly reproduces the zeromagnetization plateau attributable to the TH phase (22),but it also features the field-driven phase transition tothe one-fifth plateau pertinent to the MT phase (19)and a gradual smoothing of the magnetization curvesachieved upon rising the temperature. The numericalED and FTLM data indeed bear evidence to a legiti-macy of this effective description up to moderate tem-perature kBT/J1 ≈ 0.2, which is however nearly a halfof the temperature set as the upper bound for a reason-able description of the magnetization curves in the highlyfrustrated parameter space J2/J1 > 2. Furthermore, thespecific heat of the spin-1/2 Heisenberg octahedral chainwith the interaction ratio J2/J1 = 1.8 also displays anintriguing double-peak dependence on the magnetic fieldas exemplified in the right panel of Fig. 16. The valid-ity of the effective description of the specific heat of thespin-1/2 Heisenberg octahedral chain is again limited torelatively low temperatures kBT/J1 . 0.1, above whichthe effective monomer-dimer model fails to reproduce theprecise numerical ED and FTLM data at a quantitativelevel and provides qualitative insight only. Nevertheless,it should be mentioned that the difference in the height oftwo round maxima of the specific heat around the field-induced phase transition can be simply interpreted interms of the effective monomer-dimer lattice-gas model(26). The height of the specific-heat maximum is lowerbelow the respective field-driven phase transition, be-cause thermal excitations from the two-fold degenerate

13

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

0.2

0.1

0.5

kB T / J1 = 0.01

ED full (20 spins) MDM (20 spins)

J2 / J1 = 2.1

m

/ mS

h / J1 0 1 2 3 4 5 6

0.00

0.04

0.08

0.12

0.16 ED full (20 spins) MDM (20 spins)

0.1

0.01

kBT / J1 = 0.2

c / k

B

h / J1

0 1 2 3 4 5 60.0

0.2

0.4

0.6

0.8

1.0

0.2

0.1

0.5

kBT / J1 = 0.01

FTLM (30 spins) MDM (30 spins)

J2 / J1 = 2.1

m / mS

h / J1 0 1 2 3 4 5 6

0.00

0.04

0.08

0.12

0.16 FTLM (30 spins) MDM (30 spins)

0.1

0.01

kBT / J1 = 0.2

c / k

B

h / J1 FIG. 15. (Left panel) Magnetization curves of the spin-1/2 Heisenberg octahedral chain with J2/J1 = 2.1 obtained from theeffective monomer-dimer model (MDM) versus the ED data for 20 spins (upper panel) and the FTLM data for 30 spins (lowerpanel) of the original model (18). (Right panel) The same as in the left panel but for the specific heat. Thin black lines showanalytical results derived from the effective monomer-dimer model (26), while colored symbols refer to the ED (upper panel)or FTLM (lower panel) data.

TH state (22) are less intense than the ones from thenondegenerate MT state (19).

VI. CONCLUSION

The present article is devoted to magnetic properties ofthe spin-1/2 Heisenberg diamond and octahedral chains,which are inferred from one-dimensional lattice-gas mod-els of hard-core monomers and dimers deduced by meansof the modified localized-magnon theory. An eligibilityof the monomer-dimer lattice-gas models for elucidationof low-temperature magnetization curves and thermo-dynamic quantities of the spin-1/2 Heisenberg diamondand octahedral chains was decisively confirmed by state-of-the-art numerical calculations based on the ED andFTLM. Beside this, the ground-state phase diagrams ofthe spin-1/2 Heisenberg diamond and octahedral chainswere established by making use of exact and DMRG cal-culations.

It has been evidenced that the anticipated monomer-dimer lattice-gas model provides a proper description

of the spin-1/2 Heisenberg diamond chain when it isdriven by the interaction parameters either to the singlettetramer-dimer phase, the monomer-dimer phase or thesaturated ferromagnetic phase. Similarly, the monomer-dimer lattice-gas model captures low-temperature fea-tures of the spin-1/2 Heisenberg octahedral chain pro-vided that it is driven to the singlet tetramer-hexamerphase, the monomer-tetramer phase, the bound magnon-crystal phase or the saturated ferromagnetic phase. Thedeveloped localized-magnon theory also brings insightinto a remarkable behavior of the specific heat, whichshows a marked minimum at magnetic fields inherent todiscontinuous field-driven phase transitions that is sur-rounded by two generally asymmetric round peaks origi-nating from vigorous thermal excitations to low-lying lo-calized many-magnon eigenstates. Note furthermore thatthe previous localized-magnon theories [23, 25] valid onlyin a highly frustrated parameter region J2/J1 > 2 can berecovered from Eqs. (15) and (30) after neglecting thesecond term under the square root in the limit of suffi-ciently low temperatures and positive magnetic fields.

Let us conclude our study by presenting a few future

14

0.0 0.2 0.4 0.6 0.80.00

0.05

0.10

0.15

0.20

0.2

0.1

0.5

kBT / J1 = 0.01

ED full (20 spins) MDM (20 spins)

J2 / J1 = 1.8

m

/ mS

h / J1 0.0 0.2 0.4 0.6 0.8

0.00

0.04

0.08

0.12

ED full (20 spins) MDM (20 spins)

0.01

kB T / J1 = 0.1

J2 / J1 = 1.8

c / k

B

h / J1

0.0 0.2 0.4 0.6 0.80.00

0.05

0.10

0.15

0.20

0.2

0.1

0.5

kBT / J1 = 0.01

FTLM (30 spins) MDM (30 spins)

J2 / J1 = 1.8

m / mS

h / J1 0.0 0.2 0.4 0.6 0.8

0.00

0.04

0.08

0.12

FTLM (30 spins) MDM (30 spins)

0.01

kB T / J1 = 0.1

J2 / J1 = 1.8

c / k

B

h / J1 FIG. 16. (Left panel) Magnetization curves of the spin-1/2 Heisenberg octahedral chain with J2/J1 = 1.8 obtained from theeffective monomer-dimer model (MDM) versus the ED data for 20 spins (upper panel) and the FTLM data for 30 spins (lowerpanel) of the original model (18). (Right panel) The same as in the left panel but for the specific heat. Thin black lines showanalytical results derived from the effective monomer-dimer model (26), while colored symbols refer to the ED (upper panel)or FTLM (lower panel) data.

outlooks. We are convinced that the proposed calcula-tion scheme has opened the route to a low-temperaturemagnetic behavior of several frustrated Heisenberg spinmodels, which have at least two tensor-product groundstates with a magnetic unit spread over one and two unitcells, respectively. The mixed spin-(1,1/2) Heisenberg oc-tahedral chain affords a suitable platform for implemen-tation of the lattice-gas model of hard-core monomersand dimers, which would extend a legitimacy of the pre-vious calculations based on the lattice-gas model of hard-core monomers [27]. Note furthermore that the sug-gested calculation procedure is also compatible with re-cent extension enabling a straightforward computationof entanglement measures [28]. Most importantly, thelow-temperature physics of two-dimensional frustratedHeisenberg spin models with two tensor-product groundstates, which have magnetic unit spread over one andtwo unit cells, would be captured by a two-dimensionallattice-gas model of hard-core monomers and dimers. Fi-nally, the spin-1/2 Heisenberg orthogonal-dimer chain

[56] and the mixed-spin Heisenberg diamond chain [57]with an infinite series of fragmented quantum groundstates provide another intriguing platform for further ex-tension of the present approach, which would however re-quire consideration of one-dimensional lattice-gas modelsof hard-core monomers, dimers, trimers, tetramers, etc.

ACKNOWLEDGMENTS

J.Str. and K.K. acknowledge kind hospitality duringsummer 2021 at Max-Planck-Institut fur Physik Kom-plexer Systeme in Dresden, where a part of this workwas completed. J.Str. and K.K. acknowledge financialsupport provided by the grant The Ministry of Educa-tion, Science, Research and Sport of the Slovak Republicunder the contract No. VEGA 1/0105/20 and by thegrant of the Slovak Research and Development Agencyunder the contract No. APVV-20-0150. J.R. and J. Sch.acknowledge to DFG Grant.

15

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