Exactly solved mixed spin-(1,1/2) Ising-Heisenberg distorted diamond chain
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Exactly solved mixed spin-(1,1 /
2) Ising-Heisenberg distorted diamond chain
Bohdan Lisnyi a,b,1 , Jozef Streˇcka a,2, ∗ a Institute of Physics, Faculty of Science, P. J. ˇSaf´arik University, Park Angelinum 9, 040 01 Koˇsice, Slovak Republic b Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, 79011 Lviv, Ukraine
Abstract
The mixed spin-(1,1 /
2) Ising-Heisenberg model on a distorted diamond chain with the spin-1 nodal atoms and the spin-1 / Keywords:
Ising-Heisenberg model, distorted diamond chain, spin frustration, magnetization plateau
PACS: + q, 75.10.Hk, 75.10.Jm, 75.30.Kz, 75.40.Cx
1. Introduction
During the last few years, a considerable research interest has been devoted to the frustrated magnetism of diamondspin chains, which was initiated by several unusual magnetic features of the natural mineral azurite Cu (CO ) (OH) such as for instance an existence of the one-third magnetization plateau in a low-temperature magnetization curve[1–6]. However, it turns out that one has to employ sophisticated first-principle density-functional calculations incombination with the extensive state-of-the-art numerical calculations in order to provide a comprehensive descriptionof the overall magnetic behavior of the azurite [5, 6]. Compared to this, some exactly solved Ising-Heisenberg spinchains a ff ord a plausible quantitative description of the magnetic behavior of real spin-chain materials after modestcalculations in spite of a certain over-simplification of the physical reality [7–14].In this regard, a lot of attention has been paid to a rigorous treatment of various versions of the spin-1 / / / et al. [40] have exploredthe mixed-spin Ising diamond chain with the spin-1 nodal atoms and the spin-1 / /
2) Ising diamond chain displays peculiar magnetization plateaus, which have been later refuted by the exactcalculations based on the generalized decoration-iteration transformation [41]. Besides, we have recently furnished a ∗ Corresponding author.
Email addresses: [email protected] (Bohdan Lisnyi), [email protected] (Jozef Streˇcka) B.L. acknowledges the financial support provided by the National Scholarship Programme of the Slovak Republic for the Support of Mobilityof Students, PhD Students, University Teachers, Researchers and Artists. J.S. acknowledges the financial support provided by the grant of The Ministry of Education, Science, Research and Sport of the SlovakRepublic under the contract Nos. VEGA 1 / /
15 and VEGA 1 / /
16, as well as, by grants of the Slovak Research and Development Agencyprovided under Contract Nos. APVV-0097-12 and APVV-14-0073.
Preprint submitted to Physica A September 20, 2018 k S k +1 I I I I σ k, σ k, { J , J , J } Figure 1: A fragment from the mixed-spin Ising-Heisenberg distorted diamond chain. The nodal ( S k , S k + ) Ising spins and interstitial ( σ k , , σ k , )Heisenberg spins belonging to the k th primitive unit cell are marked. rigorous proof that the striking magnetization plateaus proposed in Ref. [40] cannot appear neither in the generalizedmixed spin-(1,1 /
2) Ising-Heisenberg diamond chain, which accounts for the single-ion anisotropy acting on the nodalspin-1 atoms [42].The main purpose of this work is to examine the ground state and basic thermodynamic properties of the mixedspin-(1,1 /
2) Ising-Heisenberg model on a distorted diamond chain, which accounts for asymmetry of the Ising cou-plings along the sides of elementary diamond plaquette distorted to parallelogram. The distorted version of themixed spin-(1,1 /
2) Ising-Heisenberg diamond chain reduces to the usual mixed spin-(1,1 /
2) Ising-Heisenberg dia-mond chain when the asymmetry of the Ising couplings along the diamond sides vanishes, while the mixed spin-(1,1 / /
2) Ising-Heisenberg distorted diamond chain will be exactly calculated within the framework of the transfer-matrix method. In particular, we will explore how a mutual competition between the geometric spin frustration andthe coupling asymmetry will a ff ect the overall magnetic behavior.The organization of this paper is as follows. The model and basic steps of the exact method will be clarified in Sec.2. The most interesting results for the ground-state phase diagrams and thermodynamic properties will be discussedin Sec. 3. Finally, the paper ends up with several concluding remarks and future outlooks mentioned in Sec. 4.
2. Model and its exact solution
Let us begin by considering the mixed spin-(1,1 /
2) Ising-Heisenberg distorted diamond chain in a presence of theexternal magnetic field. The magnetic structure of the investigated model system is schematically illustrated in Fig. 1together with its primitive unit cell. As one can see, the primitive unit cell in a shape of diamond spin cluster involvestwo nodal Ising spins S k and S k + along with two interstitial Heisenberg spins ˆ σ k , and ˆ σ k , . The total Hamiltonianfor the mixed spin-(1,1 /
2) Ising-Heisenberg distorted diamond chain in a presence of the external magnetic field readsˆ H = N X k = S k (cid:16) I ˆ σ zk , + I ˆ σ zk , + I ˆ σ zk − , + I ˆ σ zk − , (cid:17) + N X k = (cid:16) J ˆ σ xk , ˆ σ xk , + J ˆ σ yk , ˆ σ yk , + J ˆ σ zk , ˆ σ zk , (cid:17) − N X k = h (cid:16) S k + ˆ σ zk , + ˆ σ zk , (cid:17) , (1)which involves the nodal Ising spins S k = ± , σ k , i = / i = ,
2) by assumingthe periodic boundary conditions ˆ σ α , ≡ ˆ σ α N , , ˆ σ α , ≡ ˆ σ α N , ( α = x , y , z ). The interaction constants I and I label thenearest-neighbor interactions between the nodal Ising spins and interstitial Heisenberg spins along sides of the primi-tive diamond unit cell, while the coupling constants J , J and J determine the spatially anisotropic XYZ interactionbetween the nearest-neighbor interstitial Heisenberg spins from the same primitive cell. Finally, the Zeeman’s term h determines the magnetostatic energy of the nodal Ising spins and interstitial Heisenberg spins in the external magneticfield. It should be mentioned that the particular case I = I =
0) of the Hamiltonian (1) corresponds to themixed spin-(1,1 /
2) Ising-Heisenberg doubly decorated chain.2or further manipulations, it is quite advisable to rewrite the total Hamiltonian (1) as a sum over cell Hamiltoniansˆ H = N X k = ˆ H k , (2)whereas the cell Hamiltonian ˆ H k involves all the interaction terms of the k th diamond unit cellˆ H k = J ˆ σ xk , ˆ σ xk , + J ˆ σ yk , ˆ σ yk , + J ˆ σ zk , ˆ σ zk , + I (cid:16) S k ˆ σ zk , + ˆ σ zk , S k + (cid:17) + I (cid:16) S k ˆ σ zk , + ˆ σ zk , S k + (cid:17) − h (cid:16) ˆ σ zk , + ˆ σ zk , (cid:17) − h S k + S k + ) . (3)Due to the fact that the cell Hamiltonians ˆ H k commute between themselves, h ˆ H k , ˆ H n i =
0, the partition function ofthe mixed spin-(1,1 /
2) Ising-Heisenberg diamond chain can be written in this form
Z ≡
Tr exp (cid:16) − β ˆ H (cid:17) = X { S k } N Y k = Z k ( S k , S k + ) , (4)where the symbol P { S k } marks a summation over all possible spin configurations of the nodal Ising spins and Z k ( S k , S k + ) = Tr { σ k , , σ k , } exp (cid:16) − β ˆ H k (cid:17) (5)is the e ff ective Boltzmann’s factor obtained after tracing out spin degrees of freedom of two Heisenberg spins fromthe k -th primitive cell [ β = / ( k B T ), k B is the Boltzmann’s constant, T is the absolute temperature]. To proceedfurther with a calculation, one necessarily needs to evaluate the e ff ective Boltzmann’s factor Z k ( S k , S k + ) given byEq. (5). For this purpose, it is quite advisable to pass to the matrix representation of the cell Hamiltonian ˆ H k in thebasis spanned over four available states of two Heisenberg spins σ zk , and σ zk , : | ↑ , ↑i k = |↑i k , |↑i k , , | ↓ , ↓i k = |↓i k , |↓i k , , | ↑ , ↓i k = |↑i k , |↓i k , , | ↓ , ↑i k = |↓i k , |↑i k , , (6)whereas |↑i k , i and |↓i k , i denote two eigenvectors of the spin operator ˆ σ zk , i with the respective eigenvalues σ zk , i = / − /
2. After a straightforward diagonalization of the cell Hamiltonian ˆ H k one obtains the following four eigenvalues: E k , = − h S k + S k + ) + J ± s(cid:18) J − J (cid:19) + (cid:20) I + I S k + S k + ) − h (cid:21) , E k , = − h S k + S k + ) − J ± s(cid:18) J + J (cid:19) + (cid:18) I − I (cid:19) ( S k − S k + ) . (7)Now, one may simply use the eigenvalues (7) in order to calculate the Boltzmann’s factor (5) according to the relation Z k ( S k , S k + ) = " β h S k + S k + ) exp (cid:18) β J (cid:19) cosh β s(cid:18) J + J (cid:19) + (cid:18) I − I (cid:19) ( S k − S k + ) + exp (cid:18) − β J (cid:19) cosh β s(cid:18) J − J (cid:19) + (cid:20) I + I S k + S k + ) − h (cid:21) . (8)The Boltzmann’s factor (8) can be subsequently replaced through the generalized decoration-iteration transfor-mation [43–46] similarly as we have done this before [41, 42]. But since this Boltzmann’s factor is essentially atransfer matrix (see eq. (4)), we can directly apply the transfer-matrix method [47, 48]. For convenience we definethe elements V i j of the transfer matrix V as follows: V i j ≡ Z k ( S k = i − , S k + = j − = " β h i + j − exp (cid:18) β J (cid:19) cosh β s(cid:18) J + J (cid:19) + (cid:18) I − I (cid:19) ( i − j ) + exp (cid:18) − β J (cid:19) cosh β s(cid:18) J − J (cid:19) + (cid:20) I + I i + j − − h (cid:21) . (9)3he calculation of the partition function thus requires finding of the eigenvalues of the transfer matrix V = V V V V V V V V V . The eigenvalues of this matrix are given by the roots of cubic characteristic equation λ + K λ + K λ + K = , (10)where K = −
13 ( V + V + V ) , K = V V + V V + V V − V − V − V , K = V V + V V + V V − V V V − V V V . The expressions for the three transfer-matrix eigenvalues are as follows (see, e.g., [49]) λ i = | K | + √ p cos " φ + π i − , i = , , , (11)where p = (cid:16) V + V + V (cid:17) + h ( V − V ) + ( V − V ) + ( V − V ) i ,φ =
13 arccos − q p p , q = (cid:16) | K | − | K | p + K (cid:17) . As a result, the partition function Z given by Eq. (4) is determined by the transfer-matrix eigenvalues λ , λ , λ through the formula: Z = Tr V N = λ N + λ N + λ N . (12)Exact results for other thermodynamic quantities follow quite straightforwardly from the formula (12) for thepartition function Z . In the thermodynamic limit N → ∞ , the Gibbs free energy per unit cell can be evaluated fromthe formula g = − β lim N →∞ N ln Z = − β ln λ , (13)where λ = max { λ , λ , λ } is the largest transfer-matrix eigenvalue. It is easy to see from Eq. (9) that V i j > T > V is positive matrix for all non-zero temperatures.The Perron-Frobenius theorem (see, e.g., [50]) implies that the transfer matrix has a positive largest eigenvalue, whichis a simple root of the characteristic equation (10), and any other eigenvalue is strictly smaller that it in absolute value.The largest eigenvalue among the three transfer-matrix eigenvalues (11) is then given by λ = | K | + √ p cos ( φ ) . (14)The entropy s and the specific heat c per unit cell can be subsequently calculated from the formulas s = k B β ∂ g ∂β = k B ln λ − βλ ∂λ ∂β ! , c = − β ∂ s ∂β = k B β λ ∂ λ ∂β − λ ∂λ ∂β ! , whereas the total magnetization m and magnetic susceptibility χ readily follow from the relations m = − ∂ g ∂ h = βλ ∂λ ∂ h , χ = ∂ m ∂ h = β λ ∂ λ ∂ h − λ ∂λ ∂ h ! . It is quite convenient to get the derivatives ∂λ ∂β , ∂ λ ∂β , ∂λ ∂ h , ∂ λ ∂ h from Eq. (10) rather than from the final formula (14).4 . Results and discussions Now, let us proceed to a discussion of the most interesting results obtained for the mixed spin-(1,1 /
2) Ising-Heisenberg distorted diamond chain with the antiferromagnetic Ising interactions I > I >
0. To reducenumber of free parameters, we will further assume the XXZ Heisenberg interaction J = J = J ∆ , J = J , whichmay be either antiferromagnetic or ferromagnetic in character ( ∆ determines a spatial anisotropy in this interaction).Without loss of generality, we may also assume that one of two considered Ising couplings is stronger than the otherone I ≥ I and to introduce the di ff erence between both Ising coupling constants δ I = I − I >
0. Subsequently, onegets the following four-dimensional parameter space spanned over the dimensionless interaction parameters˜ J = JI , ∆ , ˜ h = hI , δ ˜ I = δ II = − I I . The parameters ˜ J and ∆ determine a relative strength of the XXZ Heisenberg interaction, while the parameter ˜ h standsfor a relative strength of the external magnetic field. Last, the parameter δ ˜ I restricted to the interval δ ˜ I ∈ [0 ,
1] hasa physical sense of the distortion parameter, because it determines a relative di ff erence between two Ising couplingconstants I and I due to the parallelogram distortion. The ground state of the mixed spin-(1,1 /
2) Ising-Heisenberg diamond chain can be trivially connected to thelowest-energy eigenstate of the cell Hamiltonian (3) obtained by taking into account all nine states of the nodal Isingspins S k and S k + entering into the respective eigenvalues (7). Depending on a mutual competition between theparameters ∆ , ˜ J > δ ˜ I and ˜ h one finds in total four di ff erent ground states: the saturated paramagnetic (SPA) state,the monomer-dimer (MD) state, the classical antiferromagnetic (AF) state, and the quantum antiferromagnetic (QAF)state given by the eigenvectors | SPA i = N Y k = | + i k ⊗ | ↑ , ↑i k , | MD i = N Y k = | + i k ⊗ √ (cid:16) | ↑ , ↓i k − | ↓ , ↑i k (cid:17) , | AF i = N Q k = |−i k ⊗ | ↑ , ↑i kN Q k = | + i k ⊗ | ↓ , ↓i k , | QAF i = N Q k = (cid:12)(cid:12)(cid:12) [ − ] k E k ⊗ (cid:16) A [ − ] k + | ↑ , ↓i k − A [ − ] k | ↓ , ↑i k (cid:17) N Q k = (cid:12)(cid:12)(cid:12) [ − ] k + E k ⊗ (cid:16) A [ − ] k | ↑ , ↓i k − A [ − ] k + | ↓ , ↑i k (cid:17) . (15)In above, the ket vector |±i k determines the state of the nodal Ising spin S k = ±
1, the symbol [ − ] k ∈ {− , + } marks thesign of the number ( − k , the spin states relevant to two Heisenberg spins from the k th primitive cell are determinedby the notation (6), and the probability amplitudes A ± are explicitly given by the expressions A ± = √ vuut ± δ ˜ I q(cid:16) ˜ J ∆ (cid:17) + δ ˜ I . (16)The eigenenergies per primitive cell that correspond to the respective ground states (15) are given as follows˜ E SPA = ˜ J + − δ ˜ I − h , ˜ E MD = − ˜ J −
12 ˜ J ∆ − ˜ h , ˜ E AF = ˜ J − + δ ˜ I , ˜ E QAF = − ˜ J − r(cid:16) ˜ J ∆ (cid:17) + δ ˜ I . (17)Let us make a few comments on two notable spin arrangements emerging within the QAF and MD ground states.Apparently, one encounters within the QAF ground state a singlet-like state of the Heisenberg spin pairs, which ischaracterized by a quantum superposition of two antiferromagnetic states | ↑ , ↓i k and | ↓ , ↑i k occurring according to Eq.(16) with two di ff erent occurrence probabilities. As a result, there appears an e ff ective staggered magnetic moment onthe Heisenberg spin pairs, which is subsequently transferred also to the nodal Ising spins provided that there is someasymmetry in both Ising couplings δ ˜ I ,
0 0.2 0.4 0.6 0.8 1.01234
0 0.1 0.20.010.020.03 J + / I = J / I = D = J / I = D = J / I = D = J / I = D = a QAFAF
MDSPA h / I d I / I J + / I = J / I = D = J / I = D = J / I = D = b QAF
MDSPA h / I d I / I Figure 2: The ground-state phase diagram in the δ ˜ I − ˜ h plane for a few typical values of the antiferromagnetic Heisenberg coupling constant ( ˜ J > ∆ ), which are consistent with two di ff erent values of the parameter ˜ J + : (a) ˜ J + =
3; (b) ˜ J + =
4. The bullet symbols • shown in Fig. 2a allocate twolimiting positions of the triple point, at which the ground states AF, QAF, and MD coexist together on assumption that ˜ J + = of the nodal Ising spins, which are consequently forced by the external magnetic field to align towards the magnetic-field direction. In this regard, the MD ground state should manifest itself in a respective magnetization curve as anintermediate plateau at one-half of the saturation magnetization.The ground-state phase diagram in the δ ˜ I − ˜ h plane can have topology of two di ff erent types depending on astrength of the parameter ˜ J + = J + / I = ˜ J (1 + ∆ ) connected to the Heisenberg interaction (see Fig. 2). The firsttype of the ground-state phase diagram involving all four available ground states SPA, MD, AF and QAF is found onassumption that ˜ J + < δ ˜ I = δ ˜ I AF | QAF ≡ − ˜ J + + ˜ J + ∆ + ∆ ) − ˜ J + ! . (18)If the Heisenberg coupling constant ˜ J + is fixed, then, the distortion parameter is restricted to the interval (cid:18) − ˜ J + / , − (cid:16) ˜ J + / (cid:17) (cid:19) at the coexistence line δ ˜ I = δ ˜ I AF | QAF as the exchange anisotropy ∆ varies from 0 to ∆ → ∞ . The ground-state phase di-agram depicted in Fig. 2a additionally contains a special triple point given by the coordinates h δ ˜ I AF | QAF , − δ ˜ I AF | QAF − ˜ J + / i ,at which the three ground states AF, QAF and MD coexist together. For the fixed value of the interaction pa-rameter ˜ J + the position of the triple point moves with the increase of parameter ∆ along the line from the point h − ˜ J + / , − ˜ J + / i at ∆ = h − ( ˜ J + / , (1 − ˜ J + / i reached in the ∆ → ∞ limit (see bulletsymbols in Fig. 2a).The second type of the ground-state phase diagram can be detected for ˜ J + ≥ ff erentground states SPA, MD and QAF (see Fig. 2b). In this particular case the phase boundary between the QAF and MDground states starts at the point with the coordinates [0 , ,
0] of the ground-state phase diagram, which corresponds to the symmetric case at zeromagnetic field, entails for ˜ J + > | FRU i = N Y k = (cid:16) |±i k or | i k (cid:17) ⊗ √ (cid:16) | ↑ , ↓i k − | ↓ , ↑i k (cid:17) , where the ket vector | i k determines the non-magnetic state of the nodal Ising spin S k =
0. Hence, the FRU groundstate has the residual entropy s res = k B ln 3 reflecting the macroscopic degeneracy 3 N , which stems from the degrees offreedom of the nodal Ising spins. It should be also mentioned that the two ground states FRU and AF coexist togetherat the point [0 ,
0] for ˜ J + =
4. 6 D = D > + I / I ) h / I = D / ( D+ ( J + / I ) + - I / I ) ] / - D/ ( D+ J + / I } h / I = + I / I - . J + / I h / I = + I / I + . J + / I - I / I + I / I I / I QAF
AF MDSPA h / I J + / I Figure 3: The ground-state phase diagram in the ˜ J + − ˜ h plane. The scale of the ˜ J + -axis is expressed in terms of the interaction ratio I / I . Solidlines determine the ground-state phase boundaries for ∆ >
0, along which the relevant mathematical formulas are also explicitly given. Brokenlines illustrate the ground-state phase boundaries for the special case ∆ =
0. The horizontal dotted line shows an imaginary continuation of thephase boundary to its intersection with the ˜ h -axis, while the oblique dotted line shows how a position of the triple coexistence point AF-QAF-MDmoves along the ˜ J + -axis. The general form of the ground-state phase diagram can be obtained in the ˜ J + − ˜ h plane when the scale of the˜ J + -axis is expressed in terms of the interaction ratio ˜ I = I / I between both Ising coupling constants (see Fig. 3).The phase boundary between the AF and QAF ground states then reads˜ J + = ˜ J + AF | QAF ≡ I + ˜ I , ∆ = ∆ − q(cid:16) − ˜ I (cid:17) + I ∆ − − ˜ I ! , ∆ , . It is noteworthy that the FRU ground state does exist in the relevant ground-state phase diagram along the line given by˜ h = J + > I = Depending on a mutual competition between the parameters ∆ , ˜ J = −| ˜ J | < δ ˜ I and ˜ h one finds in total four di ff er-ent ground states for the particular case of the ferromagnetic Heisenberg interaction: the saturated paramagnetic (SPA)state, the monomer-dimer (MD1) state, the classical antiferromagnetic (AF) state, and the quantum antiferromagnetic(QAF1) state. Two new quantum ground states MD1 and QAF1 are given by the eigenvectors | MD1 i = N Y k = | + i k ⊗ √ (cid:16) | ↑ , ↓i k + | ↓ , ↑i k (cid:17) , | QAF1 i = N Q k = (cid:12)(cid:12)(cid:12) [ − ] k E k ⊗ (cid:16) A [ − ] k + | ↑ , ↓i k + A [ − ] k | ↓ , ↑i k (cid:17) N Q k = (cid:12)(cid:12)(cid:12) [ − ] k + E k ⊗ (cid:16) A [ − ] k | ↑ , ↓i k + A [ − ] k + | ↓ , ↑i k (cid:17) , (19)whereas they di ff er from the analogous ground states MD and QAF just by the symmetric (instead of antisymmetric)quantum superposition of the antiferromagnetic states | ↑ , ↓i k and | ↓ , ↑i k of the Heisenberg spin pairs. The eigenener-gies per primitive cell, which correspond to the respective ground states (19), follow from˜ E MD1 = − ˜ J − | ˜ J | ∆ − ˜ h , ˜ E QAF1 = − ˜ J − r(cid:16) ˜ J ∆ (cid:17) + δ ˜ I . (20)The ground-state phase diagram in the δ ˜ I − ˜ h plane can have four di ff erent topologies depending on a parameter˜ J − = J − / I = | ˜ J | ( ∆ −
1) (see Fig. 4). The first type of the ground-state phase diagram (Fig. 4a) is found for ˜ J − ≤
0. Itincludes only two ground states SPA and AF. The second type of the ground-state phase diagram (Fig. 4b) is realized7
0 0.1 0.2 0.3 0.40.010.020.03
0 0.2 0.4 0.6 0.8 1.0123
0 0.2 0.4 0.6 0.8 1.0120 0.2 0.4 0.6 0.8 1.012 J - / I = J / I = - D = J / I = - D = QAF1
MD1 dSPA h / I d I / I J - / I = J / I = - D = J / I = - D = J / I = - D = QAF1
MD1 cAF SPA h / I d I / I MD1 b J - / I = D = J - / I = D = AFSPA h / I d I / I a J - / I £
0 ( D £ AFSPA h / I d I / I Figure 4: The ground-state phase diagram in the δ ˜ I − ˜ h plane for a few typical values of the ferromagnetic Heisenberg coupling constant ( ˜ J < ∆ ), which are consistent with four di ff erent topologies depending on a value of the parameter ˜ J − : (a) ˜ J − ≤
0; (b) ˜ J − = . .
0; (c) ˜ J − = .
5; (d)˜ J − = .
0. The bullet symbols • shown in Fig. 4c allocate two limiting positions of the triple point, at which the ground states AF, QAF1, and MD1coexist together on assumption that ˜ J − = . < ˜ J − ≤ / ( ∆ +
1) and it involves three di ff erent ground states: SPA, AF, and MD1. The third type of the ground-state phase diagram (Fig. 4c) involving all four available ground states SPA, MD1, AF, and QAF1 can be detected for4 / ( ∆ + < ˜ J − <
4. The AF and QAF1 ground states are separated by the phase boundary δ ˜ I = δ ˜ I AF | QAF1 ≡ − ˜ J − + ˜ J − ∆ ∆ − + ˜ J − ! . (21)If the Heisenberg coupling constant ˜ J − is fixed, then, the distortion parameter is restricted to the interval (cid:18) − (cid:16) ˜ J − / (cid:17) , − ˜ J − / , (cid:19) at the coexistence line δ ˜ I = δ ˜ I AF | QAF1 as the exchange anisotropy varies from ∆ → ∞ to ∆ →
1. The ground-statephase diagram also contains a special triple point with the coordinates h δ ˜ I AF | QAF1 , − δ ˜ I AF | QAF1 − ˜ J − / i , at which theAF, QAF1, MD1 phases coexist together (see the inset in Fig. 4c). For the fixed value of the interaction parameter˜ J − the triple point moves with the change of parameter ∆ along the line from the point h − ( ˜ J − / , (1 − ˜ J − / i at ∆ → ∞ to the point h − ˜ J − / , i at ∆ →
1. The fourth type of the ground-state phase diagram (Fig. 4d) emerges for˜ J − ≥ ff erent ground states: SPA, MD1, and QAF1. The phase boundary between the QAF1and MD1 states starts from the special point [0 , J − > | FRU1 i = N Y k = (cid:16) |±i k or | i k (cid:17) ⊗ √ (cid:16) | ↑ , ↓i k + | ↓ , ↑i k (cid:17) . The two ground states FRU1 and AF coexist together at the point [0 ,
0] for ˜ J − =
4. The FRU1 ground state has theresidual entropy s res = k B ln 3 reflecting the macroscopic degeneracy 3 N , which comes from spin degrees of freedomof the nodal Ising spins. The only di ff erence between two frustrated ground states FRU1 and FRU lies in symmetricvs. antisymmetric quantum superposition of two antiferromagnetic states | ↑ , ↓i k and | ↓ , ↑i k of the Heisenberg spinpairs.The general form of the ground-state phase diagram can be obtained in the ˜ J − − ˜ h plane when the ˜ J − -axis isexpressed in terms of the interaction ratio ˜ I = I / I (Fig. 5). Under this circumstance, the phase boundary betweenthe AF and QAF1 ground states is given by˜ J − = ˜ J − AF | QAF1 ≡ ∆ + r(cid:16) − ˜ I (cid:17) + I ∆ + + ˜ I . It should be noticed that the FRU1 ground state does exist in the relevant ground-state phase diagram along the linegiven by ˜ h = J − > I = Next, let us examine the magnetization process and other thermodynamic characteristics as a function of thetemperature and the distortion parameter. To illustrate all possible scenarios, we have selected the values of theHeisenberg coupling constants ˜ J and ∆ in order to fall into the parameter region pertinent to the ground-state phasediagrams shown in Fig. 2a and Fig. 4c involving all available ground states.The total magnetization is plotted in Fig. 6a against the magnetic field at a few di ff erent temperatures for theantiferromagnetic Heisenberg coupling ˜ J = ∆ = J + =
2) and the distortion parameter δ ˜ I = .
2, whereas therespective thermal dependences of the total magnetization at constant magnetic fields are displayed in Fig. 6b. Thezero-temperature dependence of the total magnetization depicted in Fig. 6a exhibits two abrupt magnetization jumpsclosely connected with an existence of two intermediate magnetization plateaus: the zero plateau corresponding to theAF ground state and the one-half plateau corresponding to the MD ground state. In accordance with this statement,temperature dependences of the total magnetization shown in Fig. 6b asymptotically tend towards zero, one-half orunity as temperature goes to zero. The only two exceptions from this rule are the intermediate values of the totalmagnetization m / m s = / /
4, which can be achieved in the zero-temperature asymptotic limit on assumptionthat the magnetic field is fixed exactly at the critical value corresponding to the field-induced transitions AF ↔ MD9 h / I = D / ( D- ( J - / I ) + - I / I ) ] / - D/ ( D- J - / I } + I / I ) ( D £
1) ( D > h / I = + I / I - . J - / I h / I = + I / I + . J - / I h / I = + I / I QAF1
AF MD1SPA h / I J - / I Figure 5: The ground-state phase diagram in the ˜ J − − ˜ h plane. The scale of the ˜ J − -axis is expressed in terms of the interaction ratio I / I . Solidlines determine the ground-state phase boundaries, along which the relevant mathematical formulas are also explicitly given. The oblique dottedline shows how a position of the triple coexistence point AF-QAF1-MD1 moves along the ˜ J − -axis. and MD ↔ SPA, respectively (see dotted lines in Fig. 6b). Besides, it can be observed from Fig. 6b that the totalmagnetization exhibits a relatively steep thermally-induced increase (decrease) at low enough temperatures when themagnetic field is selected slightly below (above) the critical field associated with the respective magnetization jump.The qualitatively same magnetization curve can be detected also for the other particular case δ ˜ I > δ ˜ I AF | QAF , whichdrives the zero-field ground state towards the QAF phase instead of the AF phase. The only qualitative di ff erence isthat the total magnetization normalized with respect to the saturation magnetization asymptotically reaches in a zero-temperature limit the specific value m / m s = / (2 √ ≈ . ↔ MD is chosen.
0 1 2 3 40.20.40.60.81.0 0 0.5 1.0 1.50.20.40.60.81.0 a k B T / I
0 0.01 0.1 0.2 0.3 m / m s h / I b k B T / I h / I = m / m s Figure 6: (a) The total magnetization normalized with respect to the saturation magnetization as a function of the magnetic field at a few di ff erenttemperatures for the particular case with the antiferromagnetic Heisenberg interaction ˜ J = ∆ = J + =
2) and the distortion parameter δ ˜ I = . J = ∆ = δ ˜ I = . h = . .
8, at which two di ff erent ground states coexist together. The zero-field susceptibility times temperature product ( χ T ) is presented in Fig. 7 as a function of the temperaturefor two particular cases: when the distortion parameter δ ˜ I ≤ δ ˜ I AF | QAF drives the investigated system towards theAF ground state or when the distortion parameter δ ˜ I ≥ δ ˜ I AF | QAF drives the investigated system towards the QAFground state. Obviously, the susceptibility times temperature product vanishes ( χ T →
0) as temperature goes tozero ( T →
0) regardless of a relative strength the distortion parameter δ ˜ I due to the antiferromagnetic character of10 k B T / I a d I / I
0 0.1 0.2 0.3 0.4 c k B T d I / I b k B T / I c k B T Figure 7: The zero-field susceptibility times temperature product as a function of temperature for the antiferromagnetic Heisenberg interaction˜ J = . ∆ = J + =
3) and several values of the distortion parameter: (a) δ ˜ I ≤ δ ˜ I AF | QAF ; (b) δ ˜ I ≥ δ ˜ I AF | QAF . both zero-field ground states AF and QAF. Moreover, it can be understood from Fig. 7a that the susceptibility timestemperature product increases over the whole temperature range when the distortion parameter δ ˜ I increases fromzero up to ˜ I AF | QAF . Contrary to this, the susceptibility times temperature product increases only at relatively highertemperatures upon further strengthening of the distortion parameter from ˜ I AF | QAF to 1, while the reverse trend isgenerally observed at lower temperatures (see Fig. 7b).Furthermore, let us explore an influence of the distortion parameter δ ˜ I on typical temperature dependences of thezero-field specific heat as exemplified in Fig. 8 on the particular example with the fixed value of the antiferromagneticHeisenberg interaction ˜ J = . ∆ = J + = ffi ciently small δ ˜ I . .
15. On the contrary, the more intriguing thermal dependence of the zero-field specific heat can be found forstronger parallelogram distortions δ ˜ I & .
15. Under this condition, the zero-field specific heat exhibits at least twoseparate maxima originating from diverse energy scales of two di ff erent Ising couplings, whereas quantum fluctuationsraised by XY -part of the XXZ Heisenberg coupling generally support a splitting of the emergent peaks (e.g. comparesolid and dashed lines in Fig. 8b). However, the most striking thermal variations of the zero-field specific heat canbe detected in a close vicinity of the phase boundary δ ˜ I AF | QAF between the AF and QAF ground states, where theadditional third sharp maximum emerges at low enough temperatures (see the insets in Figs. 8a and b). The novellow-temperature peak can be interpreted as the Schottky-type maximum, which relates to thermal excitations fromthe AF ground state towards the low-lying first excited QAF state or vice versa. It actually turns out that the positionof the emergent low-temperature peak moves towards lower temperatures as the distortion parameter approaches theboundary value δ ˜ I AF | QAF , whereas the height of low-temperature peak c / k B ≈ .
146 is also in an excellent accordancewith the Schottky theory for a two-level system with equal degeneracy [51, 52].To gain an overall insight, the e ff ect of parallelogram distortion on typical thermal variations of the zero-fieldspecific heat are displayed in Fig. 9 on one illustrative example of the ferromagnetic Heisenberg interaction ˜ J = − . ∆ = J − = . / ( ∆ + < ˜ J − < ff erent zero-field ground statesAF and QAF1 depending on a relative strength of the distortion parameter (see Fig. 4c). Although the temperaturedependences of the zero-field specific heat are quantitatively di ff erent, they undergo the same qualitative changes uponstrengthening of the distortion parameter. As a matter of fact, the double-peak thermal dependences of the zero-fieldspecific heat can be observed in Fig. 9 for strong enough distortion parameter δ ˜ I & .
3, whereas the remarkabletriple-peak thermal dependences with the relatively sharp low-temperature Schottky maximum do emerge when thedistortion parameter drives the investigated system close to the phase boundary between the AF and QAF1 groundstates (i.e. δ ˜ I AF | QAF1 ).At this stage, let us perform a more comprehensive analysis of the outstanding triple-peak temperature depen-dences of the zero-field specific heat, which appear due to a small deviation of the distortion parameter δ ˜ I either from11
0 0.1 0.2 0.3 0.4 0.5 0.60.10.20.30.4 0 0.1 0.2 0.3 0.4 0.5 0.60.10.20.30.4 d I / I = k B T / I a c / k B d I / I = b k B T / I c / k B Figure 8: The temperature dependences of the zero-field specific heat for the antiferromagnetic Heisenberg interaction ˜ J = . ∆ = J + =
3) andseveral values of the distortion parameter δ ˜ I : (a) δ ˜ I ≤ δ ˜ I AF | QAF = .
4; (b) δ ˜ I ≥ δ ˜ I AF | QAF . The dashed lines correspond to another particular case˜ J = ∆ = J + =
0 0.05 0.100.10.20 0.05 0.100.10.2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.10.20.30.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.10.20.30.4 d I / I = k B T / I a c / k B d I / I = b k B T / I c / k B Figure 9: The temperature dependences of the zero-field specific heat for the ferromagnetic Heisenberg interaction ˜ J = − . ∆ = J − = . δ ˜ I : (a) δ ˜ I ≤ δ ˜ I AF | QAF1 = . δ ˜ I ≥ δ ˜ I AF | QAF1 . The insets show in an enlargened scale twolow-temperature peaks emerging close to the phase boundary between the AF and QAF1 ground states. the coexistence point δ ˜ I AF | QAF of the AF and QAF ground states (Fig. 8) or the coexistence point δ ˜ I AF | QAF1 of the AFand QAF1 ground states (Fig. 9). The temperature corresponding to the maximum of the Schottky-type peak, whichoriginates from thermal excitations between the AF and QAF (QAF1) phases, is proportional to the respective energydi ff erence ˜ E AF − ˜ E QAF ( ˜ E AF − ˜ E QAF1 ) in a neighborhood of the coexistence point δ ˜ I AF | QAF ( δ ˜ I AF | QAF1 )˜ E AF − ˜ E QAF ˜ E AF − ˜ E QAF1 ≈ κ ± ( J , ∆ ) (cid:18) δ ˜ I − δ ˜ I AF (cid:12)(cid:12)(cid:12)(cid:12) QAFQAF1 (cid:19) , κ ± ( J , ∆ ) = + δ ˜ I AF (cid:12)(cid:12)(cid:12)(cid:12) QAFQAF1 r ∆ ( ∆ ± ˜ J ± + (cid:18) δ ˜ I AF (cid:12)(cid:12)(cid:12)(cid:12) QAFQAF1 (cid:19) . Clearly, the interval of values δ ˜ I in which the specific heat exhibits three separate peaks is inversely proportional tothe coe ffi cient κ ± . Another interesting observation is that the energy di ff erence ˜ E AF − ˜ E MD for ˜ J > δ ˜ I AF | QAF is the same as the energy di ff erence ˜ E AF − ˜ E MD1 for ˜ J < δ ˜ I AF | QAF1 provided that ˜ J + / + δ ˜ I AF | QAF = ˜ J − / + δ ˜ I AF | QAF1 :˜ E AF − ˜ E MD ˜ E AF − ˜ E MD1 ) = ˜ h − +
12 ˜ J ± + δ ˜ I AF (cid:12)(cid:12)(cid:12)(cid:12) QAFQAF1 + (cid:18) δ ˜ I − δ ˜ I AF (cid:12)(cid:12)(cid:12)(cid:12) QAFQAF1 (cid:19) . This means that the energy spectrum composed of three states AF, QAF, and MD in a vicinity of the coexistencepoint δ ˜ I AF | QAF is equivalent to the energy spectrum of three states AF, QAF1, and MD1 in a neighborhood of theother coexistence point δ ˜ I AF | QAF1 . For this reason, the low-temperature features of the heat capacity in a vicinity ofboth coexistence points δ ˜ I AF | QAF and δ ˜ I AF | QAF1 are equivalent provided that ˜ J + / + δ ˜ I AF | QAF = ˜ J − / + δ ˜ I AF | QAF1 . Tobe more specific, the particular cases of the specific heats shown in Fig. 8 and Fig. 9 are consistent with the followingvalues of the proportionality constants κ + ( ˜ J + , ∆ ) = .
471 and κ − ( ˜ J − , ∆ ) = . J + / + δ ˜ I AF | QAF = . J − / + δ ˜ I AF | QAF1 = . J = . ∆ = J + =
3) hasthree separate peaks in a slightly smaller interval of the distortion parameter δ ˜ I than the specific heat shown in Fig. 9for the ferromagnetic Heisenberg interaction ˜ J = − . ∆ = J − = . h / I
0 0.078 0.08 0.084 d I / I = k B T / I a c / k B h / I
0 0.07 0.08 0.09 d I / I = b k B T / I c / k B Figure 10: The semi-logarithmic plot of temperature dependences of the specific heat for the antiferromagnetic Heisenberg interaction ˜ J = . ∆ = J + = h and two di ff erent values of the distortion parameter δ ˜ I selected close to the coexistence point δ ˜ I AF | QAF = .
4: (a) δ ˜ I = . δ ˜ I = . Last but not least, let us examine temperature variations of the specific heat in a presence of non-zero externalmagnetic field. The most interesting thermal variations of the zero-field specific heat has been formerly found in aclose neighborhood of the coexistence point δ ˜ I AF | QAF between the AF and QAF ground states and hence, our primaryattention will be therefore paid to this parameter region. It can be seen from Fig. 10 that a suitable choice of thedistortion parameter and small magnetic field gives rise to a remarkable temperature dependence of the specific heatwith up to four separate maxima – the main and three additional. The three out of four peaks of the specific heat alreadyemerge at zero magnetic field. While the round maximum at the highest temperature involves thermal excitations ofdiverse physical origin, the preferred thermal excitations in between the AF and QAF states can be entirely connectedwith the Schottky-type maximum emerging at the lowest temperature. The third subtle maximum, which can beobserved in the zero-field specific heat at moderate temperatures, originates from other preferential thermal excitationsfrom two lowest-energy AF and QAF states towards low-lying excited states arising out from the FRU state. It is quiteobvious from Fig. 10 that the height of the moderate maximum is rapidly suppressed by a relatively small magneticfield, whereas the moderate maximum simultaneously splits into two less marked maxima. The outstanding splittingof the moderate peak at a relatively small magnetic field can be attributed to the Zeeman’s splitting of available spinstates of the nodal Ising spins within the highly degenerate FRU states. Finally, it is worthwhile to remember thatthe similar temperature dependence of the specific heat with four distinct peaks has been already reported for theundistorted mixed spin-(1,1 /
2) Ising–Heisenberg diamond chain ( δ ˜ I = ff ect of the uniaxial single-ion anisotropy and the magnetic field [42].13 . Conclusion In the present article we have rigorously examined the ground state and thermodynamics of the mixed spin-(1,1 / ff ects the magnetization process, susceptibility and specific heat of the mixedspin-(1,1 /
2) Ising–Heisenberg distorted diamond chain with the antiferromagnetic Ising interactions and either theantiferromagnetic or ferromagnetic XXZ Heisenberg interaction. Under this circumstances, the magnetic propertiesof the mixed spin-(1,1 /
2) Ising–Heisenberg distorted diamond chain are substantially influenced by a geometric spinfrustration.The ground-state phase diagram of the mixed spin-(1,1 /
2) Ising–Heisenberg distorted diamond chain with theantiferromagnetic (ferromagnetic) Heisenberg interaction totally consists of four di ff erent ground states: the satu-rated paramagnetic state SPA, the classical antiferromagnetic state AF, the monomer-dimer state MD (MD1) and thequantum antiferromagnetic state QAF (QAF1). The quantum ground states MD and QAF emerging for the antifer-romagnetic Heisenberg interaction di ff er from the analogous quantum ground states MD1 and QAF1 emerging forthe ferromagnetic Heisenberg interaction just by antisymmetric and symmetric quantum superposition of two an-tiferromagnetic states of the Heisenberg spin pairs, respectively. The ground-state phase diagram in the distortionparameter – magnetic field ( δ ˜ I , ˜ h ) plane can have two di ff erent topologies depending on the antiferromagnetic Heisen-berg coupling ˜ J + = ˜ J ( ∆ +
1) and four di ff erent topologies depending on the ferromagnetic Heisenberg coupling˜ J − = | ˜ J | ( ∆ − /
2) Ising–Heisenberg distorted dia-mond chain may involve at most two di ff erent intermediate plateaus at zero and one-half of the saturation magneti-zation. From this perspective, the distortion parameter does not lead to a creation of novel magnetization plateaus incomparison with the undistorted case [42]. On the other hand, the distortion parameter is responsible for a rich varietyof temperature dependences of the specific heat, which may display one, two or three anomalous low-temperaturepeaks in addition to the round maximum observable at higher temperatures. The physical origin of all observedlow-temperature peaks of the specific-heat has been clarified on the grounds of preferred thermal excitations. It isworthwhile to remark that the investigated spin system reduces to the mixed spin-(1,1 /
2) Ising–Heisenberg doublydecorated chain in the particular case I = δ ˜ I =
1) and the symmetric mixed spin-(1,1 /
2) Ising–Heisenberg dia-mond chain in the other particular case I = I ( δ ˜ I =
0) [42].
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