Breakdown of intermediate one-half magnetization plateau of spin-1/2 Ising-Heisenberg and Heisenberg branched chains at triple and Kosterlitz-Thouless critical points
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Breakdown of intermediate one-half magnetization plateau of spin-1/2Ising-Heisenberg and Heisenberg branched chains at triple and Kosterlitz-Thoulesscritical points
Katarína Karl’ová, ∗ Jozef Strečka, and Marcelo L. Lyra Institute of Physics, Faculty of Science, P. J. Šafárik University, Park Angelinum 9, 04001 Košice, Slovakia Instituto de Fisica, Universidade Federal de Alagoas, 57072-970 Maceió, AL, Brazil (Dated: August 16, 2019)The spin-1/2 Ising-Heisenberg branched chain composed of regularly alternating Ising spins andHeisenberg dimers involving an additional side branching is rigorously solved in a magnetic field bythe transfer-matrix approach. The ground-state phase diagram, the magnetization process and theconcurrence measuring a degree of bipartite entanglement within the Heisenberg dimers are exam-ined in detail. Three different ground states were found depending on a mutual interplay betweenthe magnetic field and two different coupling constants: the modulated quantum antiferromagneticphase, the quantum ferrimagnetic phase and the classical ferromagnetic phase. Two former quantumground states are manifested in zero-temperature magnetization curves as intermediate plateaus atzero and one-half of the saturation magnetization, whereas the one-half plateau disappears at a triplepoint induced by a strong enough ferromagnetic Ising coupling. The ground-state phase diagramand zero-temperature magnetization curves of the analogous spin-1/2 Heisenberg branched chainwere investigated using DMRG calculations. The latter fully quantum Heisenberg model involves,besides two gapful phases manifested as zero and one-half magnetization plateaus, gapless quantumspin-liquid phase. The intermediate one-half plateau of the spin-1/2 Heisenberg branched chainvanishes at Kosterlitz-Thouless quantum critical point between gapful and gapless quantum groundstates unlike the triple point of the spin-1/2 Ising-Heisenberg branched chain.
PACS numbers: 75.10.Jm, 75.30.Kz, 75.40.Cx, 03.65.UdKeywords: Ising-Heisenberg model, Hesienberg model, branched chain, magnetization plateaus, DMRG sim-ulations
I. INTRODUCTION
For many years one-dimensional quantum spin chainshave become of great scientific interest both from thetheoretical point of view as well as from the ex-perimental point of view.
Quantum Heisenbergspin chains can for instance exhibit many interest-ing features in a magnetization process such as mag-netization plateau, quantum spin liquid, orquasiplateau.
Among these magnetization anoma-lies, the intermediate plateaus should obey a quantiza-tion condition known as Oshikawa-Yamanaka-Affleck rule S t − m t = integer , where S t and m t is a total spin andtotal magnetization per elementary unit, respectively. Quantum ground states of the Heisenberg spin chainsensue and/or break down at quantum phase transitions,which may be however very different in character.
For instance, the intermediate 3/7-plateau of themixed spin-(1/2,5/2,1/2) Heisenberg branched chainterminates at the Kosterlitz-Thouless quantum crit-ical point. In the present paper we will examinethe ground-state phase diagram and magnetizationprocess of the related spin-1/2 Ising-Heisenberg andHeisenberg branched chains, whose magnetic structureis inspired by the heterobimetallic coordination polymer[(Tp) Fe (CN) (OCH )(bap)Cu (CH OH) · OH.H O](Tp=tris(pyrazolyl)hydroborate, bapH = 1,3-bis(amino)-2-propanol) to be further abbreviated as Fe Cu , whichincorporates the highly anisotropic trivalent Fe cations and the almost isotropic divalent Cu cations (see Fig.1). The magnetic features of the polymeric coordinationcompound Fe Cu should be described within the frame-work of the spin-1/2 XXZ Heisenberg branched chain,which cannot be solved exactly. However, the trivalentFe magnetic ions in a low-spin state ( S =1/2) possesa relatively high degree of the magnetic anisotropy dueto unquenched orbital momentum. In this regardwe will at first examine the spin-1/2 Ising-Heisenbergbranched chain by considering the highly anisotropictrivalent Fe ions as the classical Ising spins and thealmost isotropic divalent Cu cations as the quantumHeisenberg spins. This simplification allows a deriva-tion of exact results for the spin-1/2 Ising-Heisenbergbranched chain and often might be regarded as a goodapproximation of its full quantum XXZ Heisenbergcounterpart. Beside this, we will adapt density-matrix renormalization group (DMRG) method in orderto determine the ground-state phase diagram of the fullyquantum spin-1/2 Heisenberg branched chain, which willbe confronted with exact results for the simpler spin-1/2Ising-Heisenberg branched chain.The organization of this paper is as follows. Thespin-1/2 Ising-Heisenberg branched chain is introducedand solved by the use of the transfer-matrix method inSec. II. The ground-state phase diagram, magnetiza-tion curves and the concurrence between the Heisenbergdimers of the spin-1/2 Ising-Heisenberg branched chainare presented in Sec. III. The most interesting results forthe spin-1/2 Heisenberg branched chain are presented inSec. IV and finally some summarized ideas are posted inSec. V.
II. ISING-HEISENBERG BRANCHED CHAIN
Let us consider the spin-1/2 Ising-Heisenberg branchedchain schematically depicted in Fig. 2, which can bedefined through the following Hamiltonian ˆ H = N X i =1 n J h ∆ (cid:16) ˆ S x ,i ˆ S x ,i + ˆ S y ,i ˆ S y ,i (cid:17) + ˆ S z ,i ˆ S z ,i i − J (cid:16) ˆ S z ,i ˆ σ z ,i + ˆ S z ,i ˆ σ z ,i +1 + ˆ S z ,i ˆ σ z ,i (cid:17) − h (cid:16) ˆ S z ,i + ˆ S z ,i (cid:17) − h ˆ σ z ,i − h ˆ σ z ,i o , (1)where ˆ σ zi and ˆ S αi ( α = x, y, z ) denote the spatial com-ponents of the spin-1/2 operators related to the Isingand Heisenberg spins, respectively. The coupling con-stant J > stands for the antiferromagnetic Heisenberginteraction inside of dimeric Cu -Cu units from abackbone of the polymeric chain, while the coupling con-stant J > ( J < ) describes the ferromagnetic (an-tiferromagnetic) Ising-type interaction between nearest-neighbor Ising and Heisenberg spins approximating themagnetically anisotropic Fe and magnetically isotropicCu ions, respectively. Last but not least, Zeeman’sterms h j ( j = 1 , , ) are assigned to a coupling of theIsing and Heisenberg spins with an external magneticfield, N denotes the total number of unit cells. For sim-plicity, periodic boundary conditions σ ,N +1 ≡ σ , areassumed.For further convenience it is advisable to rewrite theHamiltonian (1) as a sum of the cell Hamiltonians ˆ H = N X i =1 ˆ H i , (2)where the cell Hamiltonian ˆ H i is defined by ˆ H i = J h ∆ (cid:16) ˆ S x ,i ˆ S x ,i + ˆ S y ,i ˆ S y ,i (cid:17) + ˆ S z ,i ˆ S z ,i i − J (cid:16) ˆ S z ,i ˆ σ z ,i + ˆ S z ,i ˆ σ z ,i +1 + ˆ S z ,i ˆ σ z ,i (cid:17) − h (cid:16) ˆ S z ,i + ˆ S z ,i (cid:17) − h ˆ σ z ,i − h (cid:0) ˆ σ z ,i + ˆ σ z ,i +1 (cid:1) . (3)The cell Hamiltonians ˆ H i apparently commute, i.e. [ ˆ H i , ˆ H j ] = 0 , which means that the partition functionof the spin-1/2 Ising-Heisenberg branched chain can bewritten in this form Z = X { σ } Tr e − β P i ˆ H i = X { σ z ,i } N Y i =1 X σ z ,i Tr [ S ,i ,S ,i ] e − β ˆ H i = X { σ z ,i } N Y i =1 T( σ z ,i ; σ z i +1 ) , (4) where β = 1 / ( k B T ) , k B is the Boltzmann’s factor and T is the absolute temperature. Tr [ S ,i ,S ,i ] denotes a traceover degrees of the Heisenberg dimer from the i -th unitcell, the symbol P { σ z ,i } marks a summation over all pos-sible spin configurations of the Ising spins from a back-bone chain and the expression T( σ z ,i ; σ z i +1 ) = X σ z ,i Tr [ S ,i ,S ,i ] e − β ˆ H i (5)is the effective Boltzmann’s factor obtained after tracingout spin degrees of freedom of two Heisenberg spins andthe Ising spin σ ,i at lateral branching. To proceed fur-ther with a calculation, one necessarily needs to evaluatethe effective Boltzmann’s factor given by Eq. (5). Forthis purpose, it is advisable to pass to a matrix represen-tation of the cell Hamiltonian ˆ H i in the basis spannedover four available states of two Heisenberg spins S ,i and S ,i | ↑ , ↑i i = | ↑i ,i | ↑i ,i , | ↑ , ↓i i = | ↑i ,i | ↓i ,i , | ↓ , ↑i i = | ↓i ,i | ↑i ,i , | ↓ , ↓i i = | ↓i ,i | ↓i ,i . (6)whereas | ↑i k,i and | ↓i k,i ( k = 1 , ) denote two eigenvec-tors of the spin operator ˆ S zk,i with the respective eigenval-ues S zk,i = ± / . After a straightforward diagonalizationof the cell Hamiltonian ˆ H i one obtains the following foureigenvalues E i, i = J ± J (cid:0) σ z ,i + σ z ,i + σ z ,i +1 (cid:1) ± h − h (cid:0) σ z ,i + σ z ,i +1 (cid:1) − h σ z ,i ,E i, i = − J ± q(cid:0) σ z ,i + σ z ,i − σ z ,i +1 (cid:1) +( J ∆) − h (cid:0) σ z ,i + σ z ,i +1 (cid:1) − h σ z ,i (7)and the corresponding eigenvectors | ϕ ,i i = | ↑i ,i | ↑i ,i , | ϕ ,i i = | ↓i ,i | ↓i ,i , | ϕ ,i i = c + | ↑i ,i | ↓i ,i + c − | ↑i ,i | ↓i ,i , | ϕ ,i i = c + | ↑i ,i | ↓i ,i − c − | ↑i ,i | ↓i ,i , (8)where c ± = 1 √ vuut ± J (cid:0) σ z ,i +1 − σ z ,i − σ z ,i (cid:1)q J (cid:0) σ z ,i +1 − σ z ,i − σ z ,i (cid:1) + ( J ∆) . (9)Note that the effective Boltzmann’s factor T( σ z ,i ; σ z i +1 ) given by Eq. (5) depends only on two Ising spins frombackbone of the spin chain and can be alternativelyviewed as the transfer matrix defined by T( σ z ,i ; σ z i +1 )= X σ z ,i Tr [ S ,i ,S ,i ] e − β ˆ H i = X σ z ,i = ± X j =1 e − βE ji Cu ( =1/2) S Fe ( =1/2) s FIG. 1: A part of the crystal structure of the heterobimetalic coordination polymer[(Tp) Fe (CN) (OAc)(bap)Cu (CH OH) · OH.H O] adopted according to the crystallographic data reported in Ref.30 Smaller (blue) circles determine positions of the divalent Cu + magnetic ions and larger (orange) circles determine latticeposition of trivalent Fe + magnetic ions. J J J J J J J J s i ( S i ) s i ( S i ) s i+ ( S i+ ) S i S i FIG. 2: A schematic illustration of the spin-1/2 Ising-Heisenberg branched chain. Dark (blue) circles denote lattice positionsof the Heisenberg spins and light (orange) circles denote lattice positions of the Ising spins. The notation for an analogous purespin-1/2 Heisenberg branched chain is given in round brackets whenever it differs from the spin-1/2 Ising-Heisenberg branchedchain. = 2e βh ( σ z ,i + σ z ,i +1 ) − βJ × (cid:26) e βh cosh (cid:20) β (cid:18) J σ z ,i + J σ z ,i +1 + J h (cid:19)(cid:21) + e − βh cosh (cid:20) β (cid:18) J σ z ,i + J σ z ,i +1 − J h (cid:19)(cid:21) + e βJ + βh cosh β s(cid:18) J σ z ,i − J σ z ,i +1 + J (cid:19) +( J ∆) + e βJ − βh cosh β s(cid:18) J σ z ,i − J σ z ,i +1 − J (cid:19) +( J ∆) . (10)A successive summation over states of a set of the Isingspins { σ ,i } from the backbone of a spin chain giveswithin the standard transfer-matrix approach the fol-lowing final formula for the partition function Z = X { σ ,i } N Y i =1 T( σ z ,i ; σ z i +1 ) = Tr T N = λ N + + λ N − , (11)which depends on two eigenvalues of the transfer matrix(10) λ ± = T + T ± s(cid:18) T − T (cid:19) + T T . (12) Here, the expressions T i ( i = 1 , , , ) denote specific el-ements of the transfer matrix (10) for the following par-ticular spin states of the Ising spins σ ,i and σ ,i +1 T = T (cid:18) , (cid:19) , T = T (cid:18) − , − (cid:19) ,T = T (cid:18) , − (cid:19) , T = T (cid:18) − , (cid:19) . (13)At this stage, the exact result for the partition function(11) can be used to obtain the Gibbs free energy, whichis given in the thermodynamic limit only by the largesttransfer-matrix eigenvalue G = − k B T lim N →∞ N ln Z = − k B T ln λ + . (14)One can subsequently obtain local magnetizations (andconsequently the total magnetization) by differentiatingthe Gibbs free energy (14) with respect to local magneticfields m , = 12 h ˆ S z ,i + ˆ S z ,i i = − N ∂G∂h ,m = 12 h ˆ σ z ,i + ˆ σ z ,i +1 i = − N ∂G∂h ,m = h ˆ σ z ,i i = − N ∂G∂h ,m t = 14 (2 m + m + m ) . (15)To bring insight into a degree of bipartite entanglementinside of the Heisenberg dimers one may take advantageof the concurrence, which can be expressed in terms ofthe local magnetization and the respective pair correla-tion functions. To this end, one can perform relevantdifferentiation of the Gibbs free energy (14) with respectto spatial components of the coupling constant in order tocalculate respective spatial components of the pair corre-lation function between the nearest-neighbor Heisenbergspins according to the formulas C zz = h ˆ S z ,i ˆ S z ,i i = − ∂ ln Z N ∂ ( βJ ) ,C xx = h ˆ S x ,i ˆ S x ,i i = − ∂ ln Z N ∂ ( βJ ∆) . (16)The concurrence can be then calculated from the exactresults for the pair correlation functions (16) and the lo-cal magnetization (15) according to the formula C = max
0; 4 | C xx | − s(cid:18)
14 + C zz (cid:19) − m . (17) III. THE MOST INTERESTING RESULTS
In what follows, we will consider the particular caseof the spin-1/2 Ising-Heisenberg branched chain with theisotropic ( ∆ = 1 ) antiferromagnetic Heisenberg interac-tion
J > , which will henceforth serve as an energy unitwhen defining dimensionless interaction parameters J /J and h/J measuring a relative strength of the couplingconstants and magnetic field, respectively. For simplicity,we will further assume that all particular local magneticfields are the same h = h = h = h what corresponds toassuming equal Landé g -factors of Cu and Fe mag-netic ions. By comparing energies of all lowest-energyeigenstates one can obtain the ground-state phase dia-gram of the spin-1/2 Ising-Heisenberg branched chain asdepicted in Fig. 3(a) in J /J − h/J plane. Solid lines inFig. 3(a) denote discontinuous field-induced phase tran-sitions, which split the overall parameter space into threeregions labeled as I (I’), II and III. The microscopic char-acter of the relevant ground states is schematically shownin Fig. 4 and the corresponding eigenvectors are givenby | I , I ′ i = N/ Y i =1 (cid:8)(cid:2) a + | ↓i S , i − | ↑i S , i − − a − | ↑i S , i − | ↓i S , i − (cid:3) |↑i σ , i − | ↑i σ , i − + (cid:2) ( a − | ↓i S , i | ↑i S , i − a + | ↑i S , i | ↓i S , i (cid:3) | ↓i σ , i | ↓i σ , i (cid:9) , | II i = N Y i =1 (cid:2) b + | ↓i S ,i | ↑i S ,i − b − | ↑i S ,i | ↓i S ,i (cid:3) | ↑i σ ,i | ↑i σ ,i , | III i = N Y i =1 | ↑i S ,i | ↑i S ,i | ↑i σ ,i | ↑i σ ,i . (18)The respective probability amplitudes are defined as a ± = 1 √ vuut ± J q(cid:0) J (cid:1) + ( J ∆) (19)and b ± = 1 √ vuut ± J q(cid:0) J (cid:1) + ( J ∆) . (20)The ground state I (I’) can be viewed as the modulatedantiferromagnetic phase with a twofold breaking of trans-lational symmetry, which involves a singlet-like state ofthe Heisenberg dimers and up-up-down-down spin ar-rangements of the Ising spins. Note furthermore that theground states | I i and | I ′ i emergent for the antiferromag-netic ( J < ) and ferromagnetic ( J > ) Ising coupling differ from one another just by a relative orientation ofthe singlet-like state of the Heisenberg dimers with re-spect to its surrounding Ising spins. The ground state | II i has character of the quantum ferrimagnetic phasewith other singlet-like state of the Heisenberg dimers ac-companied with the fully polarized Ising spins. It is note-worthy that these ground states have obvious quantumfeatures as exemplified by nonzero concurrence serving asa measure of bipartite entanglement within the Heisen-berg dimers. Finally, the third ground state | III i is classi-cal ferromagnetic phase with fully polarized Ising as wellas Heisenberg spins. The ground-state phase boundaries[solid lines in Fig. 3(a)] are given by the following exactprescriptions -2 -1 0 1 20.00.51.01.52.0 C h / J (a) J / J I’I IIIII t m m m m , m , m , m t (b) h / Jk B T/J0.002J /J = -1.50.0 0.2 0.4 0.6 0.80.00.10.20.30.40.5 m , m , m , m t (c) h / JJ /J = 0.5 m t m m m B T/J0.002 0.0 0.2 0.4 0.60.00.10.20.30.40.5 m , m , m , m t (d) h / JJ /J = 1.5 0.05k B T/J0.002 m t m m m FIG. 3: (a) The ground-state phase diagram of the spin-1/2 Ising-Heisenberg branched chain in the J /J − h/J plane sup-plemented with a density plot of the concurrence calculated for the Heisenberg dimers; (b)-(d) magnetic-field dependencies oflocal and total magnetizations defined according to Eq. (15) at two different temperatures and three values of the interactionratio: (b) J /J = − . ; (c) J /J = 0 . ; (d) J /J = 1 . .
1. phase boundary | I i ( | I ′ i )/ | II i : hJ = 12 s(cid:18) J J (cid:19) + 1 − s(cid:18) J J (cid:19) + 1 , (21)2. phase boundary | I ′ i / | III i : hJ = 14 − J J + 14 s(cid:18) J J (cid:19) + 1 , (22)3. phase boundary | II i / | III i : hJ = 12 − J J + 12 s(cid:18) J J (cid:19) + 1 . (23)The local and total magnetizations are plotted in Fig.3(b) against the external magnetic field for the specialcase J /J = − . and two different temperatures. Thelocal magnetization m , defined as the mean magnetiza-tion of the Heisenberg dimers exhibits only zero magne-tization plateau, because the Heisenberg dimers do notcontribute to the total magnetization up to a saturationfield due to their singlet-like character in the phases | I i and | II i (see Fig. 4). The local magnetizations m and I ( J <0) IIIIII’ >0)( J FIG. 4: A schematic illustration of spin arrangments withinall possible ground states of the spin-1/2 Ising-Heisenbergbranched chain. Two arrows within a single site denotequantum superposition of both spin states, whereas larger(smaller) arrow refers to a spin state emergent with a greater(smaller) occurrence probability. m , which refer to the magnetization of the Ising spinswithin the main chain ( m ) and side branching ( m ), re-spectively, are zero only at low enough magnetic fieldsdue to their up-up-down-down spin alignment realizedwithin the phase | I i . However, the total magnetizationexhibits a discontinuous magnetization jump due to afield-driven phase transition from zero plateau (the phase | I i ) towards the one-half plateau (the phase | II i ), whichrelates to a reorientation of the Ising spins (i.e. the lo-cal magnetizatons m and m ) towards the magneticfield. It should be nevertheless remarked that at finitetemperatures there is no true magnetization plateau andjump neither in Fig. 3(b), nor in all other figures, be-cause the actual magnetization jump and plateau existat zero temperature only. It is evident from Fig. 3(b)that the magnetization curve at low enough tempera-ture k B T /J = 0 . shows a steep but continuous risewith the magnetic field, while an increase of temperaturegenerally causes a gradual smoothing of the magnetiza-tion curve. The second discontinuous field-driven phasetransition appears at a saturation field, at which a spinreorientation of the Heisenberg spins takes place.The local and total magnetizations are depicted in Fig.3(c) as a function of the magnetic field for another valueof the interaction ratio J /J = 0 . . In this parame-ter space the phases | I ′ i , | II i and | III i can be realizedas the respective ground states depending on a relativesize of the magnetic field. The only change with respectto the aforementioned particular case lies in a changeof relative orientation of the nearest-neighbor Ising andHeisenberg spins, which relates to different character ofthe Ising coupling constant. At very low magnetic fields,the phase | I ′ i is realized as the relevant ground state,whose microscopic character implies zero contributionof all local magnetizations to the total magnetization.Above the first critical field h c /J ≈ . the phase | II i becomes the ground state with zero contribution of thelocal magnetization m of the Heisenberg dimers and sat-urated values of the local magnetizations m and m ofthe Ising spins. It is obvious from Fig. 3(c) that tem-perature k B T /J = 0 . is high enough to destroy zero-magnetization plateau of the local magnetization m and m .Last but not least, we have investigated the local andtotal magnetizations in the parameter space supportingonly the phases | I ′ i and | III i as the respective groundstates. To support this statement, the local and totalmagnetizations are plotted in Fig. 3(d) against the ex-ternal magnetic field for the special case J /J = 1 . . Thephase | I ′ i is realized as the ground state below the criticalfield h c /J ≈ . , while the classical fully polarized fer-romagnetic phase | III i becomes the ground state abovethis critical field. All local magnetizations of the Isingand Heisenberg spins behave alike in this particular caseand thus, they cannot be discerned within the displayedfigure.To bring a deeper insight into a degree of bipartiteentanglement between the nearest-neighbor Heisenbergspins (dimers) we will comprehensively examine the con-currence as a function of the magnetic field and temper-ature in three different cuts of the parameter space. Itis obvious from the eigenvectors (18) that the Heisen-berg dimers are in the ground states | I i ( | I ′ i ) and | II i quantum-mechanically entangled, whereas the concur-rence characterizing bipartite entanglement achieves in a zero-temperature limit the following asymptotic values C I = C I ′ = 1 q(cid:0) J J (cid:1) + 1 ,C II = 1 q(cid:0) J J (cid:1) + 1 . (24)It follows from the formulas (24) that Heisenberg dimersare more strongly entangled in the ferrimagnetic phase | II i than in the modulated antiferromagnetic phase | I i ( | I ′ i ) for the same value of the interaction ratio J /J .The field dependence of the concurrence is displayed inFig. 5(a) for the interaction ratio J /J = − . and sev-eral values of temperature. The low-temperature asymp-totes ( k B T /J = 0 . ) of the concurrence can be under-stood from the ground-state phase diagram [Fig. 3(a)]and the formulas (24), which imply existence of threedifferent ground states | I i , | II i and | III i depending on arelative size of the magnetic field. It follows from Fig.5(a) that the concurrence is kept constant at low enoughtemperatures and then it shows abrupt changes in a vicin-ity of the critical fields associated with the magnetiza-tion jumps. The non-zero value of the concurrence up tothe second critical field h c /J = 2 . at sufficiently lowtemperatures ( k B T /J = 0 . ) proves quantum characterof the phases | I i and | II i , while the zero concurrence athigher magnetic fields h/J > . confirms classical char-acter of the phase | III i . Interestingly, the bipartite entan-glement within the Heisenberg dimers is approximatelytwo-times stronger in the quantum ferrimagnetic phase | II i than in the quantum antiferromagnetic phase | I i forthis choice of the interaction constants [see Fig. 5(a)].An increase of temperature causes a gradual smoothingof the concurrence, which is successively suppressed bythermal fluctuations above both quantum ground states | I i and | II i and contrarily reinforced above the classicalground state | III i .Typical temperature dependencies of the concurrencein the same cut of the parameter space J /J = − . are depicted in Fig. 5(b) for several values of the mag-netic field. It is evident from Fig. 5(b) that the con-currence mostly monotonically decreases with increas-ing temperature though it may also show a more strik-ing nonmonotonous temperature dependence, specificallyslightly below the saturation field. Indeed, the con-currence shows at the saturation field h c /J = 2 . agradual thermally-induced decline starting from the zero-temperature asymptotic value C ≈ . due to a coexis-tence of the phases | II i and | III i , while it exhibits a vig-orous thermally-induced decline (rise) just below (justabove) of the saturation field h/J = 2 . ( h/J = 2 . ) ow-ing to thermal excitations to the classical ferromagnetic(quantum ferrimagnetic) phase | III i ( | II i ).The concurrence is plotted in Fig. 6(a) against themagnetic field at fixed value of the interaction ratio J /J = 0 . and a few different temperatures. Underthis condition, the concurrence exhibits stepwise changes /J = -1.5 k B T/J C (a) h / J /J = -1.5 (b) k B T/J C FIG. 5: (a) The concurrence as a function of the magnetic field for the interaction ratio J /J = − . and several values oftemperature; (b) Temperature variations of the concurrence for the interaction ratio J /J = − . and several values of magneticfield. /J = 0.5 k B T/J C (a) h / J 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0 h / J 0.0 0.6 0.7 1.0 2.0J /J = 0.5 (b) k B T/J C FIG. 6: (a) The concurrence as a function of the magnetic field for the interaction ratio J /J = 0 . and several values oftemperature; (b) Temperature variations of the concurrence for the interaction ratio J /J = 0 . and several magnetic fields. close to the critical fields h c /J ≈ . and h c /J ≈ . .Apparently, a small rise of temperature can invoke in-crease of the concurrence, specifically, the concurrenceat zero magnetic field is higher at moderate temperature k B T /J = 0 . than at very low temperature k B T /J = 0 . on account of thermal excitations from the less entangledphase | I ′ i towards the more entangled phase | II i . Tosupport this statement, temperature dependence of theconcurrence is depicted in Fig. 6(b) for the interactionratio J /J = 0 . and several values of the magnetic field.The displayed temperature dependence of the concur-rence at zero magnetic field indeed corroborates a tran-sient strengthening of the bipartite entanglement withinthe range of moderate temperatures ( k B T /J ≤ . ),which is successively followed by a relatively steep de-crease at higher temperatures. The concurrence thusstarts from its highest possible value for J /J = 0 . ina range of moderate magnetic fields h/J ∈ (0 .
11; 0 . ,which stabilize the phase | II i in concordance with the ground-state phase diagram shown in Fig. 3(a). Unlikethis, the concurrence starts from zero above the satu-ration field in agreement with the classical character ofthe phase | III i [c.f. Fig. 6(a)], but afterwards it showsa marked temperature-induced rise until a round maxi-mum is reached that is successively followed by a steepdecrease upon increasing of temperature. It is worthwhileto remark that the concurrence approaches zero close toa threshold temperature k B T /J ∼ = 0 . , above which itequals zero independently of the magnetic field.Last but not least, we have examined field and tem-perature dependencies of the concurrence for the interac-tion ratio J /J = 1 . , which is consistent with only twoground states | I ′ i and | III i in accordance with the estab-lished ground-state phase diagram [see Fig. 3(a)]. In thisparticular case, one observes an abrupt fall of the con-currence at the saturation field h c /J ≈ . , which deter-mines a field-driven phase transition from the phase | I ′ i towards the phase | III i . It is apparent from Fig. 7(a) that /J = 1.5 k B T/J C (a) h / J 0.0 0.2 0.4 0.6 0.8 1.00.00.10.20.30.4 h / J 0.0 0.3 0.4 1.0 2.0J /J = 1.5 (b) k B T/J C FIG. 7: (a) The concurrence as a function of the magnetic field for the interaction ratio J /J = 1 . and several values oftemperature; (b) Temperature variations of the concurrence for the interaction ratio J /J = 1 . and several magnetic fields. the concurrence is gradually suppressed upon strength-ening of the magnetic field. The most peculiar temper-ature dependencies of the concurrence can be detectedwhen the magnetic field is selected sufficientlty close butslightly below the saturation field (e.g. h c /J = 0 . for J /J = 1 . in Fig. 7(b)). Under this condition, the con-currence shows at very low temperature a steep declineuntil it reaches a local minimum, then it passes througha round local maximum emergent at moderate tempera-tures until it finally completely vanishes at the thresholdtemperature k B T t / | J | = ≈ . . IV. HEISENBERG BRANCHED CHAIN
Next, let us consider the analogous but purely quan-tum spin-1/2 Heisenberg branched chain (see Fig. 2),which can be defined through the following Hamiltonian ˆ H = N X i =1 h J ˆ S ,i · ˆ S ,i − J (cid:16) ˆ S ,i · ˆ S ,i + ˆ S ,i · ˆ S ,i +1 + ˆ S ,i · ˆ S ,i (cid:17) − h X j =1 ˆ S zj,i . (25)Here, ˆ S j,i ( j = 1 , , , ) are standard spin-1/2 opera-tors assigned to four magnetic ions from the i th unit cell,the coupling constant J > stands for the antiferromag-netic interaction within the dimeric Cu -Cu units ofthe main chain, the coupling constant J > ( J < )stands for ferromagnetic (antiferromagnetic) interactionbetween Cu and Fe ions (see Fig. 1 and 2). The Zee-man’s term h accounts for the external magnetic field, N denotes the total number of unit cells and N t = 4 N is thetotal number of spins. For simplicity, periodic boundaryconditions are also assumed.To obtain the ground-state phase diagram and mag-netization process of the spin-1/2 Heisenberg branched chain (25) we have performed density-matrix renormal-ization group (DMRG) simulations by adapting the sub-routine from the Algorithms and Libraries for PhysicsSimulations (ALPS) project. It should be mentionedthat the DMRG data were obtained for the spin-1/2Heisenberg branched chain with N = 24 , and unitcells (i.e. with the total number of spins N T = 96 , and 192), whereas adequate numerical accuracy wasachieved through 12 sweeps at targeted system size whenincreasing the number of kept states up to 1500 duringthe final sweeps.The magnetic-field dependence of the total magnetiza-tion is displayed in Fig. 8 for several values of interac-tion ratio J /J . If the coupling constant J < is an-tiferromagnetic, the total magnetization of the spin-1/2Heisenberg branched chain first displays a zero magne-tization plateau, which ends up at field-driven quantumphase transition towards a quantum spin liquid termi-nating at second field-driven quantum phase transition(QPT) towards the one-half magnetization plateau. Theintermediate one-half plateau breaks down at third field-driven QPT when the investigated spin chain reentersquantum spin-liquid regime, which terminates at fourthfield-driven QPT emergent at a saturation field. Forthe ferromagnetic coupling constant J > , the totalmagnetization successively exhibits a tiny zero magne-tization plateau, quantum spin liquid, one-half magne-tization plateau and quantum spin liquid up to a rel-atively strong ferromagnetic interaction J /J ≈ , atwhich the intermediate one-half plateau vanishes fromthe magnetization curve. It can be seen from Fig. 8that a width of one-half plateau becomes narrower withincreasing of the interaction ratio J /J regardless of asign of the coupling constant J . Contrary to this, zeromagnetization plateau extends over a wider range of themagnetic fields upon strengthening of the antiferromag-netic coupling constant J < , while it almost remainsunchanged with respective of a relative size of the ferro- m t / m s (a) h / J J / J = -2.0 m t / m s (b) h / JJ / J = 3.0 FIG. 8: The magnetic-field dependence of the total magnetization of the spin-1/2 Heisenberg branched chain for several valuesof the interaction ratio: (a) J /J = − . , − . , − . , − . ; (b) J /J = 3 . , . , . , . . Stepwise curves display DMRG datafor a finite-size chain with N = 24 unit cells, while smooth curves are an extrapolation to thermodynamic limit N → ∞ . magnetic coupling constant J > .The magnetic-field dependence of the local and to-tal magnetizations of the spin-1/2 Heisenberg branchedchain are plotted in Fig. 9 at zero temperature for severalvalues of the relative size of the antiferromagnetic cou-pling constant J /J . It is apparent from Fig. 9 that thelocal magnetizations m , m and m ( h S ,i i , h S ,i i and h S ,i i ) are positive albeit not yet fully saturated in a fullrange of the magnetic fields, while the local magnetiza-tion m ( h S ,i i ) changes its sign. It should be stressedthat the local magnetization m achieves within the one-half plateau almost saturated value, while there is evidenta quantum reduction in all other local magnetizations dueto a presence of quantum fluctuations. It is noteworthythat all local magnetizations generally differ from oneanother. However, the local magnetizations may becomeidentical as for instance the local magnetizations m and m for the special value of interaction ratio J /J = − . in a full range of the magnetic fields [see Fig. 9(b)], oras the local magnetizations m and m for the couplingratio J /J = − . within the one-half plateau [see Fig.9(c)].The magnetic-field dependencies of local and totalmagnetizations for the spin-1/2 Heisenberg branchedchain with ferromagnetic coupling constant J > aredepicted in Fig. 10. It turns out that all local magnetiza-tions are positive in a full range of the magnetic fields ex-cept the local magnetization m , which may be orientedin opposite to the external magnetic field (negative) [seeFig. 10(a) and (b)]. It is quite curious that the localmagnetization m is almost saturated at weak ferromag-netic coupling constant J /J & [see Fig. 10(a) and Fig.10(b)], which means that it does not principally matterwhether the spins S ,i are described by the notion of clas-sically Ising or fully quantum Heisenberg spins. Hence,it follows that the spin-1/2 Heisenberg branched chainshould resemble in this particular limit a magnetic behav- ior of the spin-1/2 Ising-Heisenberg branched chain. Itshould be pointed out, moreover, that the one-half mag-netization plateau of the spin-1/2 Heisenberg branchedchain generally shrinks upon strengthening of the interac-tion ratio J /J . Owing to this fact, the one-half plateauemerges just within a very narrow range of the magneticfields and it becomes almost indiscernible within the usedscale as depicted for instance in Fig. 10(d) for the inter-action ratio J /J = 3 .Let us examine a breakdown of the intermediate one-half magnetization plateau of the spin-1/2 Heisenbergbranched chain in a somewhat more detail. A width ofthe magnetization sector, which corresponds to the inter-mediate one-half plateau, is plotted in Fig. 11(a) againsta reciprocal value of the total number of spins N t for afew different values of the interaction ratio J /J . It isquite obvious from Fig. 11(a) that a spin gap associatedwith existence of the intermediate one-half plateau closesonly very gradually upon strengthening of the interac-tion ratio J /J . A proper finite-size analysis implies thatthe one-half magnetization plateau (spin gap) still per-sists in the thermodynamic limit for the interaction ratio J /J = 3 . and . , while it completely vanishes abovea quantum critical point emergent close the interactionratio J /J . . [see Fig. 11(a)]. The data extrapo-lated for upper and lower critical fields of the intermedi-ate one-half plateau, which are displayed in Fig. 11(b) forthe interaction ratio J /J = 4 . , are in accordance withthis statement. Moreover, an exponentially slow suppres-sion of a spin gap suggests that the intermediate one-halfmagnetization plateau terminates at a quantum criticalpoint of Kosterlitz-Thouless type quite similarly as re-cently reported for a quantum critical point of the mixedspin-(1/2,5/2,1/2) Heisenberg branched chain. Whilethe magnetization curve does not bear any clear evidenceof this type of quantum criticality [see Fig. 11(c)], thesusceptibility should display a pronounced dip at the rel-0 m t m m m m (a) h / J J / J = -0.5 t m m m = m m (b) h / J J / J = -1.0 m t m m m m (c) h / J J / J = -1.5 m t m m m m (d) h / J J / J = -2.0 FIG. 9: The magnetic-field dependence of the local and total magnetization of the spin-1/2 Heisenberg branched chain forseveral values of the interaction ratio: (a) J /J = − . ; (b) J /J = − . ; (c) J /J = − . ; (d) J /J = − . . Stepwise curvesdisplay DMRG data for a finite-size chain with N = 24 unit cells, while smooth curves were obtained from an extrapolation tothermodynamic limit N → ∞ . evant quantum critical point and zero value within themagnetic-field range corresponding to the intermediateone-half plateau [see Fig. 11(d)].To confirm all aforedescribed results we have plottedin Fig. 12(a) the ground-state phase diagram of the spin-1/2 Heisenberg branched chain in the plane J /J − h/J .The ground-state phase diagram totally involves zero andone-half plateau, quantum spin liquid and fully polarizedferromagnetic phase. The ground state related to zeroplateau becomes narrower upon weakening of the antifer-romagnetic coupling constant J < , while it holds verylow (but nonzero) value for all positive values of the fer-romagnetic coupling constant J > . Contrary to this,the gapped phase related to the one-half magnetizationplateau becomes narrower upon strengthening of the fer-romagnetic coupling constant until it completely vanishesabove a quantum critical point located around the inter-action ratio J /J . . , at which gapful one-half plateauphase coexist together with the gapless spin-liquid phase.Finally, we have plotted in Fig. 12(b) the ground-statephase diagrams of the spin-1/2 Heisenberg branchedchain and the spin-1/2 Ising-Heisenberg branched chain in order to compare phase boundaries of both studiedsystems. It follows from Fig. 12(b) that the intermedi-ate one-half magnetization plateau of the spin-1/2 Ising-Heisenberg branched chain is suppressed by a quantumspin liquid. Moreover, the phase related to zero mag-netization plateau of the spin-1/2 Heisenberg branchedchain is realized only at very low magnetic fields in com-parison with zero magnetization plateau of the spin-1/2Ising-Heisenberg branched chain. On the other hand,the phase transition between one-half plateau and sat-uration is achieved almost at the same magnetic fieldfor both models within the interval of interaction ra-tio J /J < . . The most significant discrepancy isthat the intermediate one-half plateau of the spin-1/2Ising-Heisenberg branched chain ends up at a triple point( J /J ≈ . ), while the one-half plateau of the spin-1/2Heisenberg branched chain diminishes at the Kosterlitz-Thouless quantum critical point ( J /J . . ), at whichphases related to the one-half plateau and the quantumspin liquid coexist together.1 m t m m m m (a) h / JJ / J = 0.5 0.0 0.2 0.4 0.6 0.80.00.10.20.30.40.5 m m t m m m m (b) h / J J / J = 1.0 m t m m m m (c) h / J J / J = 2.0 m m m m h / J J / J = 3.0 FIG. 10: The magnetic-field dependence of the local and total magnetization of the spin-1/2 Heisenberg branched chain forseveral values of the interaction ratio: (a) J /J = − . ; (b) J /J = 1 . ; (c) J /J = 2 . ; (d) J /J = 3 . . Stepwise curvesdisplay DMRG data for a finite-size chain with N = 24 unit cells, while smooth curves are an extrapolation to thermodynamiclimit N → ∞ . V. CONCLUSION
In the present work, we have examined the ground-state phase diagram, magnetization curves and concur-rence of the spin-1/2 Ising Heisenberg branched chain bythe use of the transfer-matrix method. Besides this, wehave provided the numerical simulation in order to obtainthe ground-state phase diagram, local and total magne-tization of the spin-1/2 Heisenberg branched chain. Wehave found three different ground states in the spin-1/2Ising-Heisenberg branched chain depending on a mutualinterplay between the magnetic field and two differentcoupling constants. The modulated quantum antifer- romagnetic phase manifests itself in a zero-temperaturemagnetization process as zero plateau, the quantum ferri-magnetic phase as the intermediate one-half plateau andthe classical ferromagnetic phase is trivial fully polarizedstate. Besides zero and one-half plateau we have dis-covered a gapless quantum spin-liquid phase in a mag-netization process of the spin-1/2 Heisenberg branchedchain. The most interesting finding presented in this pa-per was that while the one-half plateau of the spin-1/2Ising-Heisenberg branched chain terminates at a triplepoint, the one-half plateau of the spin-1/2 Heisenbergbranched chain ends up at Kosterlitz-Thouless quantumcritical point. ∗ Electronic address: [email protected] T. Kuramoto, J. Phys. Soc. Jpn. , 1762 (1998). S. Yamanoto, T. Sakai, J. Phys.: Condens. Matter ,5175 (1999). T. Sakai, S. Yamanoto, Phys. Rev. B , 4053 (1999). N. B. Ivanov, Phys. Rev. B , 3217 (2000). A. Honecker, F. Mila, M. Troyer, Eur. Phys. J. B. , 227(2000). S. Yamamoto, T. Sakai, Phys. Rev. B , 3795 (2000). T. Sakai, S. Yamamoto, Phys. Rev. B , 214403 (2002). A.S.F. Tenório, R.R. Montenegro-Filho, M.D. Coutinho-Filho, J. Phys. Condens. Matter , 506003 (2011). / J N t = 96, 144, 192 3.0 3.5 4.0 4.25 4.5 h / J (a) 1 / N t t = 96, 144, 192J / J = 4.0 h c / J (b) 1 / N t N t = 96 N t = 144 N t m t / m s (c) h / J J / J = 4.0 N t = 96 N t = 144 t (d) h / JJ / J = 4.0 FIG. 11: (a) A width of the intermediate one-half plateau versus a reciprocal value of the total number of spins N t , for a fewdifferent values of the interaction ratio J /J ; (b) Upper and lower magnetic fields of the magnetization sector, which might beresponsible for intermediate one-half plateau against a reciprocal value of the total number of spins N t for the interaction ratio J /J = 4 . ; (c) The total magnetization as a function of the magnetic field for the interaction ratios J /J = 3 . and J /J = 4 . .Stepwise curves are magnetization data obtained from DMRG simulation of finite-size chains with the total number of spins N t = 96 and 144 (i.e. N = 24 and unit cells), while smooth curves are an extrapolation to thermodynamic limit N t → ∞ ;(d) The susceptibility as a function of the magnetic field for the interaction ratios J /J = 3 . and J /J = 4 . as obtained fromDMRG simulation of finite-size chains with the total number of spins N t = 96 and 144 (i.e. N = 24 and unit cells). K. Karľová, J. Strečka, Physica B , 494 (2018). J. S. Miller, M. Drillon, Magnetism: Molecules to Materi-als, Wiley-VCH, Weinheim, 2001. M. Hagiwara, K. Minami, Y. Narumi, K. Tatani, K. Kindo,J. Phys. Soc. Jpn. , 2209 (1998). K. Kopinga, A.M.C. Tinus, W.J.M. de Jonge, Phys. Rev.B , 4685 (1982). G. Kamienriarz, R. Matysiak, P. Gegenwart, H. Aoki, A.Ochiai, J. Magn. Magn. Mater. , 353 (2005). H. Kühne, H.-H. Klauss, S. Grossjohann, W. Brenig, F.J.Litterst, A.P. Reyes, P.L. Kuhns, M.M. Turnbull, C.P.Landee, Phys. Rev. B , 045110 (2009). M. Jeong, H.M. Rønnow, Phys. Rev. B , 180409(R)(2015). T. Chakraborty, H. Singh, D. Chaudhuri, H. Jeevan, P.Gegenwart, Ch. Mitra, J. Magn. Magn. Mater. , 101(2017). N. Maeshima, M. Hagiwara, Y. Narumi, K. Kindo, T.C.Kobayashi, K. Okunishi, J. Phys.: Condens. Matter ,3607 (2003). I. Maruyama, S. Miyahara, J. Phys. Soc. Jpn. , 123703 (2018). K. Morita, T. Sugimoto, S. Sotu, T. Tohyama, Phys. Rev.B , 014412 (2018). L.S. Wu, S.E. Nikitin, Z. Wang, W. Zhu, C.D. Batista,A.M. Tsvelik, A.M. Samarakoon, D.A. Tennant, M.Brando, L. Vasylechko, M. Frontzek, A.T. Savici, G.Sala, G. Ehlers, A.D. Christianson, M.D. Lumotsen, A.Podlesnyak, Nature Communications , 698 (2019). N. Avalishvili, G.I. Japaridze, G.L. Rossini, Phys. Rev. B , 205159 (2019). W.M. da Silva, R.R. Montenegro-Filho, Phys. Rev. B ,214419 (2017). S. Bellucci, V. Ohanyan, O. Rojas, Eur. Phys. Lett. ,47012 (2014). V. Ohanyan, O. Rojas, J. Strečka, S. Bellucci, Phys. Rev.B , 214423 (2015). M. Oshikawa, M. Yamanaka, I. Affleck, Phys. Rev. Lett. , 1984 (1997). K. Karľová, J. Strečka, T. Verkholyak, preprint arxiv:1904.02889. S. Ejima, T. Yagamuchi, F.H.L. Essler, F. Lange, Y. Ohta, -2 -1 0 1 2 3 4 50.00.51.01.5 spin liquid - p l a t ea u h / J (a) J / J s p i n li qu i d saturation -2 -1 0 1 2 3 4 50.00.51.01.5 h / J (b) J / J FIG. 12: (a) The ground-state phase diagram of the spin-1/2 Heisenberg branched chain in the h/J − J /J plane; (b) Acomparison between the ground-state phase diagrams of the spin-1/2 Heisenberg branched chain (solid lines) and the spin-1/2Ising-Heisenberg branched chain (broken lines).H. Feshke, SciPost. Phys. , 059 (2018). L.M. Veríssimo, M.S.S. Pereira, J. Strečka, M.L. Lyra,Phys. Rev. B , 134408 (2019). G.-H. Liu, L.-J. Kong, J.-Y. Dou, Solid State Commun. , 10 (2015). L.-C. Kang, X. Chen, H.-S. Wang, Y.-Z. Li, Y. Song, J.-L.Zuo, X.-Z. You, Inorg. Chem. , 9275 (2010). L.J. de Jongh, A.R. Miedema, Adv. Phys. , 1 (1974). R.L. Carlin, Magnetochemistry, Springer, Berlin, 1986. O. Kahn, Molecular Magnetism, New York 1993. W.P. Wolf, Braz. J. Phys.
794 (2000). J. Strečka, M. Jaščur, M. Hagiwara, K. Minami, Y.Narumi, K. Kindo, Phys. Rev. B , 024459 (2005). W. Van den Heuvel, L.F. Chibotaru, Phys. Rev. B ,174436 (2010). S. Sahoo, J.-P. Sutter, S. Ramasesha, J. Stat. Phys. ,181 (2012). J. Strečka, M. Hagiwara, Y. Han, T. Kida, Z. Honda, M.Ikeda, Condens. Matter Phys. , 43002 (2012). F. Souza, M.L. Lyra, J. Strečka, M.S.S. Pererira, J. Magn.Magn. Mater. , 423 (2019). J. Torrico, J. Strečka, M. Hagiwara, O. Rojas, S.M. de Souza, Y. Han, Z. Honda, M.L. Lyra, J. Magn. Magn.Mater. , 368 (2018). T. Verkholyak, J. Strečka, Phys. Rev. B , 144410 (2016). H. A. Kramers, G. H. Wannier, Phys. Rev. , (1941) 252. W. K. Wooters, Phys. Rev. Lett. , 2245 (1998). L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Rev. Mod.Phys. , 517 (2008). R. Horodecki, P. Horodecki, M. Horodecki, and K.Horodecki, Rev. Mod. Phys. , 865 (2009). B. Bauer, L. D. Carr, H.G. Evertz, A. Feiguin, J. Freire,S. Fuchs, L. Gamper, J. Gukelberger, E. Gull, S. Guertleret al., J. Stat. Mech. (2001) P05001.