Geometrically frustrated Cairo pentagonal lattice stripe with Ising and Heisenberg exchange interactions
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Geometrically frustrated Cairo pentagonal lattice stripewith Ising and Heisenberg exchange interactions
F. C. Rodrigues, S. M. de Souza and Onofre Rojas Departamento de Física, Universidade Federal de Lavras, CP 3037, 37200-000, Lavras-MG,Brazil
Abstract
Motivated by the recent discoveries of some compounds such as the Bi Fe O which crystallizes in an orthorhombic crystal structure with the Fe ions, andiron-based oxyfluoride Bi Fe O F compounds following the pattern of Cairopentagonal structure, among some other compounds. We propose a model forone stripe of the Cairo pentagonal Ising-Heisenberg lattice, one of the edgesof a pentagon is different, and this edge will be associated with a Heisenbergexchange interaction, while the Ising exchange interactions will associate theother edges. We study the phase transition at zero temperature, illustratingfive phases: a ferromagnetic phase (FM), a dimer antiferromagnetic (DAF), aplaquette antiferromagnetic (PAF), a typical antiferromagnetic (AFM) and apeculiar frustrated phase (FRU) where two types of frustrated states with thesame energy coexist. To obtain the partition function of this model, we use thetransfer matrix approach and following the eight vertex model notation. Usingthis result we discuss the specific heat, internal energy and entropy as a functionof the temperature, and we can observe some unexpected behavior in the low-temperature limit, such as anomalous double peak in specific heat due to theexistence of three phase (FRU, PAF(AFM) and FM) transitions occurring in aclose region to each other. Consequently, the low-lying energy thermal excitationgenerates this double anomalous peak, and we also discuss the internal energy atthe low temperature limit, where this double peak curve occurs. Some propertiesof our result were compared with two dimensional Cairo pentagonal lattices, aswell as orthogonal dimer plaquette Ising-Heisenberg chain. Keywords:
Pentagonal stripe; Ising-Heisenberg model; Geometric spinfrustration
1. Introduction
In the past few years, the investigation in the Cairo pentagonal lattice hasdrawn much attention to researchers in condensed matter physics. The first ∗ email: ors@dfi.ufla.br Preprint submitted to Elsevier September 4, 2018 odel proposed with the Cairo pentagonal Ising lattice structure was publishedin 2002 by Urumov[1], where the author studied as a purely theoretical physicsproblem, years later motivating a significant impact in pentagonal lattice investi-gation. More recently, the geometric frustration of Cairo pentagonal Ising modelalso was studied in more detail in reference [2]. By the year of 2009 Ressoucheet al.[3] identified a compound Bi Fe O which crystallizes in an orthorhombicstructure with the Fe ions, forming a first analogue with a magnetic Cairopentagonal tessellation, where the Heisenberg exchange interaction describesthe couplings with a good approximation. Furthermore, Ralko [4] studied thephase diagram of Cairo Pentagonal XXZ with spin-1/2 under a magnetic field,discussing the zero and finite temperature properties, using the stochastic seriesexpansion and the cluster mean-field theory approach. Next, Rousochatzakiset al.[5] performed an extensive investigation both analytically and numerically,for the antiferromagnetic Heisenberg model on the Cairo pentagonal lattice.Following, Pchelkina and Streltsov[6] investigated the electronic structure andmagnetic properties of compound Bi Fe O , forming a Cairo pentagonal lat-tice with strong geometric frustration. Besides, Abakumov et al.[7] reported anew crystal structure and magnetism of the iron-based oxyfluoride Bi Fe O F ,and this compound also exhibits a Cairo pentagonal structure. Later, Isoda etal.[8] also studied the magnetic phase diagram under a magnetic field[8] as wellas its magnetization process[9] of the spin-1/2 Heisenberg antiferromagnetic onthe Cairo pentagonal lattice. More recently, a novel compound Bi Fe − x Cr x O ( x = 0 . , , . ) has been synthesized using a soft chemistry technique followedby a solid-state reaction in Ar[10], which is a highly homogeneous mullite-typesolid.There are even more new compounds studied with the same structure, suchthe 2D crystals SnX (X = S, Se, or Te)[11] which have been investigated usinga first-principle calculation. Another investigation was carried out by Chainaniand Sheshadri [12] for the Ising model with Cairo pentagonal pattern with anearest-neighbor antiferromagnetic coupling. Finally, we can still comment onthe new penta-graphene, recently discovered by Zhang et al.[13], although thiscompound is a non magnetic one, the penta-graphene pattern follows exactlythe same Cairo pentagonal tessellation.On the other hand, the Cairo pentagonal lattice Ising model[1, 2] is thedual of Shastry-Sutherland lattice Ising model[14]. Motivated by the Shastry-Sutherland lattice, Ivanov proposed a quasi one-dimensional Heisenberg modelcalled as orthogonal dimer plaquette chain[15]. Certainly, the Cairo pentagonalchain can be viewed as a decorated orthogonal dimer chain[16, 17], where in ourcase the Ising spin would be considered as decorated spins. However, we can-not use the decoration transformation approach[21] naively to map the Cairopentagonal chain into an orthogonal dimer chain, because we have quantumspins instead of classical spins (required condition to apply decoration transfor-mation technique). Although there is a proposal for quantum spin decorationtransformation approach[18], this transformation is exact only for isolated deco-rations, applying to a quantum spin lattice model would be just an approximatemapping. 2he outline of this work is as follows. In sec. 2, we present the pentagonalIsing-Heisenberg chain, and we discuss the phase diagram at zero temperature.In sec. 3, we present the details to obtain the free energy calculation, and insec. 4, we discuss some physical quantities obtained from free energy, such asthe entropy, specific heat and internal energy. Finally, in sec. 5 we summarizeour results and draw our conclusions.
2. Cairo pentagonal Ising-Heisenberg lattice stripe
Motivated by the comments given in the introduction we consider a stripeof the Cairo pentagonal lattice or decorated orthogonal dimer plaquette chainwith Ising-Heisenberg coupling as schematically depicted in fig. 1. J ∆ ∆ ′ J ′ J J σ a,i σ b,i σ c,i σ d,i σ d,i − s ,i s ,i s ,i s ,i s ,i +1 s ,i +1 Figure 1: Schematic representation of the Cairo pentagonal Ising-Heisenberg lattice stripe, σ represents Heisenberg spins and s represents the Ising spins. The dashed rectangle representsan unit cell. Let us define that the Hamiltonian for a Cairo pentagonal chain by H = N X i =1 (cid:0) H abi + H cdi,i +1 (cid:1) , (1)where N is the number of cells and assuming periodic boundary condition. Thus,let us call as "elementary cell" ab -dimer and cd -dimer, which are described by H abi and H cdi,i +1 respectively.Therefore, each block Hamiltonian become H abi = − J ( σ a,i , σ b,i ) ∆ − J ( s ,i + s ,i ) σ za,i + − J ( s ,i + s ,i ) σ zb,i , (2) H cdi,i +1 = − J ′ ( σ c,i , σ d,i ) ∆ ′ − J ( s ,i + s ,i ) σ zc,i + − J ( s ,i +1 + s ,i +1 ) σ zd,i , (3)here σ αγ,i are the spin operators (with α = { x, y, z } ) at site i for particles γ = { a, b, c, d } , for detail see fig.1. The Ising spin exchange interaction parameter is3enoted by J , whereas J ( J ′ ) represents the Heisenberg exchange interactionand ∆ ( ∆ ′ ) means the anisotropic exchange interaction between Heisenbergspins for ab -dimer ( cd -dimer) respectively. Whereas for ab -dimer J ( σ a,i , σ b,i ) ∆ is defined by J ( σ a,i , σ b,i ) ∆ ≡ J ( σ xa,i σ xb,i + σ ya,i σ yb,i ) + ∆ σ za,i σ zb,i , (4)and for cd -dimer J ′ ( σ a,i , σ b,i ) ∆ ′ is defined analogously to eq.(4). To study the phase diagram at zero temperature, we need to describe theground-state energy per unit cell. One elementary cell is composed by one dimerand bonded to 4 Ising spins. Each unit cell can be composed by two elementarycells: one ab -dimer and one cd -dimer, both dimers are bonded by 2 Ising spins.In fig.1 is illustrated one possible unit cell represented by a dashed rectangle.The ground-state energy for each elementary cell could be described schemat-ically using the fancy notations by ab − dimer −→ | s s i ∼ h s s i and (5) cd − dimer −→ | s s i∼h s s i , (6)where s i correspond to the Ising spins, while the fancy symbols i ∼ h and i∼h denotefour possible states as a function of { s i } . Here, we use the spin subindex justfor convenience, which cannot be confused with a more explicit spins notationin the Hamiltonian (2) and (3). Therefore, the eigenstates | s s i ∼ h s s i of ab -dimersare conveniently expressed using four additional fancy notations, which are rep-resented as follows | i ÷ h i = | ++ i , (7) | i ≏ h i = (cid:0) − sin( φ ) | + − i + cos( φ ) | − + i (cid:1) , (8) | i ≎ h i = (cid:0) cos( φ ) | + − i + sin( φ ) | − + i (cid:1) , (9) | i − h i = | −− i , (10)where φ = arctan (cid:0) JJ ( s − s − s + s ) (cid:1) , with − π φ π .The first eigenstate | ++ i of the ab -dimer (7) is denoted by | i ÷ h i , which is linkedto spins { s , s , s , s } , thats why we use this fancy notation. Similarly, theeigenstate (10) is denoted by | i − h i which represents the states | −− i linked to thesame set of spins { s , s , s , s } . Whereas, | i ≏ h i denotes eq.(8) some kind of "anti-symmetric" state for φ > , and | i ≎ h i represents (9) some kind of "symmetric"state for φ > , although for φ < this affirmation exchanges.It is worth to mention that the states | i ÷ h i and | i − h i are independent of Isingspins { s , s , s , s } , whereas states | i ≏ h i and | i ≎ h i depends of φ which subsequentlydepends of Ising spins { s , s , s , s } . 4hereas the corresponding energy eigenvalues for ab -dimer are given by i ÷ h ǫ = − ∆4 + J ( s + s + s + s ) , (11) i ≏ h ǫ = ∆4 + q J ( s + s − s − s ) + J , (12) i ≎ h ǫ = ∆4 − q J ( s + s − s − s ) + J , (13) i − h ǫ = − ∆4 − J ( s + s + s + s ) . (14)Now using the eight-vertex model notation[1, 2, 19] and the fancy notations(7-10), we define six different states, explicitly including the Ising spins thatconnect the elementary cells by | u i = | ++ i ∼ h ++ i , | u i = | − + i ∼ h + − i , | u i = | − + i ∼ h − + i , | u i = | ++ i ∼ h −− i , | u i = | − + i ∼ h ++ i , | u i = | ++ i ∼ h + − i . (15)The reason why we choose this notation could be more evident in the nextsection.To obtain all 16 possible states, we can use in relation (15) the verticaland spin inversion symmetry, to recover 10 remaining states. Whereas eachcorresponding elementary cell ( | u i i ) energy is defined by ǫ i , which have thefollowing properties ǫ = ǫ and ǫ = ǫ = ǫ = ǫ . Note that each | u i i represents symbolically four states given by (7-10).Analogously, we can obtain the corresponding states | s s i∼h s s i , by the samerelation to that (11-14), consequently the eigenvalues are expressed merely bysubstituting J → J ′ , ∆ → ∆ ′ and { s , s , s , s } → { s , s , s , s } .Thus, the energy eigenvalues become i÷h ǫ ′ = − ∆ ′ + J ( s + s + s + s ) , (16) i ≏ h ǫ ′ = ∆ ′ + q J ( s + s − s − s ) + J ′ , (17) i ≎ h ǫ ′ = ∆ ′ − q J ( s + s − s − s ) + J ′ , (18) i−h ǫ ′ = − ∆ ′ − J ( s + s + s + s ) . (19)Substituting φ → φ ′ , the corresponding eigenstates read as follows | i÷h i = | + + i , (20) | i ≏ h i = − sin( φ ′ ) | + −i + cos( φ ′ ) | − + i , (21) | i ≎ h i = cos( φ ′ ) | + −i + sin( φ ′ ) | − + i , (22) | i−h i = | − −i , (23)where φ ′ = arctan (cid:0) J ′ J ( s + s − s − s ) (cid:1) , with − π φ ′ π .Similarly, the corresponding elementary states | u j i → | ¯ u j i can be obtainedby rotating in π/ all 16 states. Then the first rotated state becomes | ¯ u i = | ++ i∼h ++ i , thus all other states could be similarly obtained.5fter defining each elementary cell, now we can construct the unit cell statesby | u j i ⊗ | ¯ u k i . Once more, let us use the eight-vertex model notation[1, 2] tosimplify the eigenstate of the unit cell, | v i i = | s s i ∼ h s s i∼h s ′ s ′ i . (24)Here becomes useful the fancy notation, because it relates the unit cell structureclearly. The unit cell state denoted by | v i i are closely related with eight-vertexmodel notation for Ising spins { s , s , s ′ , s ′ } , here again we denote the Isingspins only for convenience by s ′ = s and s ′ = s . Thus, the eq.(24) can beexpressed as follows | v i = | ++ i ∼ h s s i∼h ++ i , | v i = | − + i ∼ h s s i∼h + − i , | v i = | − + i ∼ h s s i∼h − + i , | v i = | ++ i ∼ h s s i∼h −− i , | v i = | − + i ∼ h s s i∼h ++ i , | v i = | ++ i ∼ h s s i∼h + − i . (25)It is worth mentioning that the unit cell convention allows expressing theeigenstates, this means that the edge of a unit cell must always connect the spinsthat share two neighboring unit cells. Thus, to satisfy the number of particlesper unit cell, the leftmost and rightmost Ising spins must be shared by two unitcells, then each shared particle contributes with a half "particle" in the unitcell, as described by the fancy notations.Certainly, each unit cell state | v i i represents symbolically the × × possible states, the most relevant states are given by (25) and the remainingconfigurations can be obtained using horizontal symmetry and spin inversionsymmetry.Nevertheless, the rotational symmetry and the vertical symmetry are notallowed, because | i ∼ h i∼h i 6 = | i∼h i ∼ h i , this means that the local chiral symmetry isbroken in each unit cell, although the global chiral symmetry is preserved.Using the previous result, we can study the phase diagram of the ground-state energy for the Cairo pentagonal chain per unit cell, thus we obtain E FM = − J − ∆2 , (26) E AFM = J − q J + 4 J , (27) E PAF = − J − q J + 4 J , (28) E DAF =2 J − ∆2 , (29) E FRU = ∆ − | J | − q J + 4 J , (30)where we consider for simplicity J ′ = J and ∆ ′ = ∆ .Thus the system exhibits five states, whose ground-states can be expressed6y | F M i = N Y i =1 | ++ i ÷ h ++ i÷h ++ i i or N Y i =1 | −− i − h | −− i−h | −− i i , (31) | DAF i = N Y i =1 | ++ i − h ++ i−h ++ i i , or N Y i =1 | −− i ÷ h | −− i÷h | −− i i , (32) | P AF i = N/ Y i =1 | ++ i ÷ h ++ i ≎ h −− i − h −− i ≎ h ++ i i , (33) | AF M i = N/ Y i =1 | ++ i − h ++ i ≎ h −− i ÷ h −− i ≎ h ++ i i . (34)The Cairo pentagonal Ising-Heisenberg chain can be described by the Hamil-tonian (1), which exhibits five states (31-34): where we found a ferromagnetic( F M ) phase; three types of antiferromagnetic phase, a dimer antiferromagnetic(DAF) phase, a plaquette antiferromagnetic (PAF) phase and one antiferromag-netic (AFM); Surely, the states (31-34) satisfy the spin inversion symmetry, allIsing and Heisenberg spins inversion leaves the system invariant.The other state corresponding to the energy (30) is frustrated (
F RU ), rep-resented symbolically by | F RU i = N Q i =1 | τ i − τ i − i ≎ h − τ i − τ i i ≎ h τ i τ i i i , → Frustration type I, N Q i =1 | − τ i τ i i ≎ h − τ i τ i i ≎ h − τ i +1 τ i +1 i i , → Frustration type II , (35)where τ i ( τ i,i +1 ) can take independently ± in each unit cell. We can recog-nize a frustrated state of type I is degenerate in N states, since for each unitcell there are 2 degrees freedom (35). Similarly, for the frustration of type IIbecomes N possible configurations (states) (35). Therefore, in total we have × N degenerate states. Notice that frustration type I and II cannot be mixed,because the linking spins of each unit cells are incompatible. Thus, we find aresidual entropy S = k B ln(2 × N ) /N = k B ln(2) . It is worth remembering thatthe factor that multiplies N corresponds to two type of frustrations, but inthermodynamic limit this factor becomes irrelevant, this peculiar property isunusual for frustrated systems.In fig.2a is illustrated the phase diagram ∆ against J , for fixed J = 1 , wherewe observe all five phases. The boundary between F M and
P AF is given by ∆ = − J + p J + 1 , analogously the interface between P AF and
F RU isdescribed by ∆ = 1 − J , similarly the boundary between DAF and
AF M is limited by the curve ∆ = 2 J + p J + 1 , whereas the boundary between AF M and
F RU is described by ∆ = 1 + 2 J . In fig.2b is depicted anotherphase diagram J versus ∆ for a fixed parameter J = 1 . Illustrating once7 igure 2: Ground-state phase diagram, where is illustrated a ferromagnetic phase ( F M ), aplaquette antiferromagnetic (
P AF ), a dimer antiferromagnetic (
DAF ) and a frustrated (FRU)phase. (a) In plane J − ∆ , for fixed J = 1 . (b) In plane J − ∆ , for fixed J = 1 . again the previous phases displayed in fig.2a, whose boundary between F M and
P AF ( AF M ) is described by the curve ∆ = − √ J , and similarlythe interface between P AF ( AF M ) and F RU is given by ∆ = − | J | .It is worth to mention that, the energy degeneracy in the boundary of DAF and
F M per unit cell, each dimers (Heisenberg spins) contributes with con-figurations ( i ÷ h , i − h ) and 4 Ising spins with configurations, thus the residualentropy is S = k B ln(2 N × N ) /N = 6 k B ln(2) . There is a point for J = 0 , J = 1 and ∆ = 1 where all phases coexist which is a highly frustrated phase,each dimers (Heisenberg spins) contributes with the triplet state ( i ÷ h , i − h , i ≎ h ) and 4Ising spins ( ) whose residual entropy is S = k B ln(3 N × N ) /N = 2 k B ln(12) .Using a similar reasoning in fig.2a, the curve surrounding the frustratedregion, becomes a frustrated curve with a residual entropy S = k B ln(3) , whereasthe curve limiting the boundary between F M ( DAF ) and P AF ( AF M ) has aresidual entropy S = k B ln(2) .At first glance, in fig.2a we can observe that for ∆ > we could have aferromagnetic coupling, so we should not expect a frustrated state in this regionbecause the spins are aligned parallel to the z -axis. However, we observe a frus-trated region for ∆ > , because the Heisenberg spins have projections on the xy components which contributes with −| J | / in eq.(30), thus generating a geomet-ric frustration of quantum origin. Certainly, a quantum geometric frustrationeffect vanishes according J → becoming a classical geometric frustration.Now let us compare the phase diagram of fig.2a with that two-dimensionalCairo pentagonal Ising model[2] which exhibits a ferrimagnetic state, for detailssee fig.4 of reference [2]. In the stripe of Cairo pentagonal lattice, there is noferrimagnetic phase, in principle the DAF phase should be the responsible stateto generate the ferrimagnetic phase in the two-dimensional lattice. It is easyto recognize the top and bottom particles will be sharing with neighboring unitcells, then, we will have a non-zero magnetization per unit cell, so generating aferrimagnetic phase in two-dimensional lattice model. While the arise of
AF M phase will not be allowed in two-dimensional lattice, because the sharing particle8pin will not be equivalent between unit cells, also a similar property forbids
P AF in the two-dimensional lattice model. In a nutshell, the
AF M and
P AF phases only emerge in a one-dimensional pentagonal chain.We can view the Cairo pentagonal chain as a decorated orthogonal dimerchain[16, 17], where in the Cairo pentagonal chain the Ising spin would be con-sidered as a decorated spin. Therefore, we can compare the fig.2 with fig.3 ofreference [16], and we observe that both figures are somewhat similar, partic-ularly the phase boundaries. Although there is a difference between them, theorthogonal dimer chain does not exhibit a frustrated phase region[16], unless inthe phase boundaries.
3. Thermodynamics of the model
The partition function of a Cairo pentagonal Ising-Heisenberg stripe can beobtained through the transfer matrix technique[19].The Boltzmann factor for an ab -dimer elementary cell is given by w ( s ,i , s ,i , s ,i , s .i ) = tr ab (cid:16) e − βH abi,i (cid:17) , (36)where β = 1 /k B T , with k B being the Boltzmann’s constant and T is the absolutetemperature.Whereas for a cd -dimer the Boltzmann factor is expressed by ¯ w ( s ,i +1 , s ,i , s ,i , s .i +1 ) = tr cd (cid:16) e − βH cdi,i +1 (cid:17) . (37)The best way to perform the trace is to diagonalize the Hamiltonian for ab -dimerand cd -dimer.Using the standard 8-vertex model notation[19] as successfully used in two-dimensional spin lattice model[1, 2], we can express the Boltzmann factors for ab -dimer as follows ω = w (+ , + , + , +) = z (cid:0) x + x − (cid:1) + y + y − z , (38) ω = w (+ , − , + , − ) = 2 z + y + y − z , (39) ω = w (+ , + , − , − ) = ω , (40) ω = w (+ , − , − , +) = 2 z + y + y − z , (41) ω = w (+ , + , + , − ) = z (cid:0) x + x − (cid:1) + y + y − z , (42) ω = w (+ , + , − , +) = ω , (43) ω = w (+ , − , + , +) = ω , (44) ω = w ( − , + , + , +) = ω , (45)9here x = e βJ / , y = e βJ/ and z = e β ∆ / , we also define the followingexponents y = e β √ J + J / and y = e β √ J +4 J / just to simplify our notation.Analogously, the Boltzmann factors for the cd -dimer are expressed in a sim-ilar way to the ab -dimer. Therefore, we have ¯ ω = ¯ w (+ , + , + , +) = z ′ (cid:0) x + x − (cid:1) + y ′ + y ′− z ′ , (46) ¯ ω = ¯ w (+ , − , + , − ) =2 z ′ + y ′ + y ′− z ′ , (47) ¯ ω = ¯ w (+ , − , − , +) =2 z ′ + y ′ + y ′ − z ′ , (48) ¯ ω = ¯ w (+ , + , + , − ) = z ′ (cid:0) x + x − (cid:1) + y ′ + y ′ − z ′ , (49)where y ′ = e βJ ′ / and z ′ = e β ∆ ′ / , we define also the following exponents y ′ = e β √ J ′ + J / and y ′ = e β √ J ′ +4 J / . Moreover, we also have the followingrelations: ¯ ω = ¯ ω and ¯ ω = ¯ ω = ¯ ω = ¯ ω .To study the thermodynamics of the Cairo pentagonal Ising-Heisenbergchain, we observe that the Hamiltonian of each unit cell commutes betweenthem. Consequently, the partition function could be written as the product ofBoltzmann factors corresponding to the unit cells, Z N = tr N Y i =1 e − β ( H abi,i + H cdi,i +1 ) ! . (50)Therefore, the partition function (50) can be obtained using the transfermatrix approach, whose transfer matrix elements become T ( s , s , s ′ , s ′ ) = X s ,s w ( s , s , s , s )¯ w ( s ′ , s , s , s ′ ) . (51)The matrices w and ¯ w corresponding to ab -dimer and cd -dimer respectively, canbe expressed explicitly by w = ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω , ¯ w = ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω ¯ ω . (52)It is worth mentioning that the matrices w and ¯ w are fully symmetric ma-trices.Thus, the transfer matrix T becomes T = w ¯ w = τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ τ , (53)10here τ i are the elements of transfer matrix, which are denoted following theeight-vertex model similar to that denoted in reference [1, 2]. Then τ i areexpressed by τ = ω ¯ ω + ω ¯ ω + 2 ω ¯ ω , (54) τ = ω ¯ ω + ω ¯ ω + 2 ω ¯ ω , (55) τ = ω ¯ ω + ω ¯ ω + 2 ω ¯ ω , (56) τ = ω ¯ ω + ω ¯ ω + 2 ω ¯ ω , (57) τ = ( ω + ω ) ¯ ω + ω (¯ ω + ¯ ω ) , (58) τ = ( ω + ω ) ¯ ω + ω (¯ ω + ¯ ω ) , (59)and we can also verify that τ = τ , τ = τ and τ = τ .Note that the transfer matrix T is a non-symmetric matrix, because in gen-eral τ = τ , despite each w and ¯ w are perfectly symmetric.Thus, using the transfer matrix approach the partition function (50), can bewritten as Z N = tr h ( w ¯ w ) N i = tr h T N i . (60)The eigenvalues of transfer matrix, can be obtained from det ( T − λ ) = 0 , whichresults in, (cid:0) λ − a λ + a (cid:1) ( λ − τ + τ ) ( λ − τ + τ ) = 0 , (61)where the coefficients of a quadratic equation are given by a = ( τ + τ + τ + τ ) , (62) a = ( τ + τ ) ( τ + τ ) − τ τ . (63)Consequently, the eigenvalues of the transfer matrix can be expressed asfollows λ = τ − τ = 0 , (64) λ = τ − τ , (65) λ ± = τ + τ + τ + τ ± √ ( τ − τ − τ + τ ) +16 τ τ . (66)Although, the transfer matrix T is a non-symmetric one, we can observe thatall eigenvalues are obviously real functions. Besides, one can readily identify,there is a largest eigenvalue positively defined λ + , because all τ i are positivereal numbers following the relations (54-59).In thermodynamic limit, the free energy per unit cell depends of the thelargest eigenvalue of the transfer matrix, which is expressed by f = − β ln ( λ + ) . (67)Using the free energy, we are able to obtain several thermodynamics quan-tities. 11 . Physical quantities In what follows we will discuss the entropy ( S = − ∂f∂T ) property of the Cairopentagonal chain, illustrating the regions where the model exhibits a frustratedsector as well as the different antiferromagnetic phases found in the previoussection. Figure 3: (a) Density plot of entropy as a function of J and ∆ , assuming fixed J = 1 and T = 0 . . (b) Density plot of entropy as a function of J against ∆ for fixed J = 1 . and T = 0 . . In fig.3a, we illustrate the density plot of the entropy as a function of J and ∆ for a fixed parameter J = 1 and in the low-temperature limit T = 0 . ,darker regions correspond to higher entropies. Thus, we can readily verify theevidence of a frustrated region (FRU), with residual entropy S → k B ln(2) ,and the boundary of this region is also a frustrated state with residual entropy S → k B ln(3) . Furthermore, there is a frustration curve in the interface of FMand DAF (PAF and AFM) both with residual entropy provided by S → k B ln(2) .The darkest region corresponds to J = 0 , ∆ = 1 and J = 1 , which correspondsto a trivial frustrated phase composed by uncoupled ab -dimers and cd -dimers, sothere are 4 Ising spins per unit cell and 2 triplet state (dimers) per unit cell, asdiscussed previously then the residual entropy leads to S → k B ln(12) ≈ . .Similarly, in fig.3b we illustrate the density plot of entropy in the plane J against ∆ , in the low-temperature limit T = 0 . and for fixed parameter J = 1 . Our results exhibit once again the presence of a frustrated region ofthe model (see fig.2). The darkest region corresponds to J = 0 , thus, the Cairopentagonal chain reduces to a pure Cairo pentagonal Ising chain with residualentropy S = 4 k B ln(2) ≈ . , in this region we have an equivalent frustrationto that found in reference [2]. 12 igure 4: (a) Entropy as a function of the temperature, for fixed values of J = 1 and ∆ = − . , for a range of values of J = { . , . , . , . , . } in a logarithmic scale intemperature. (b) Specific heat as a function of temperature for a same set of parameters tothe case (a). (c) Internal energy as a function of temperature for a same set of parameters tothe case of (a), but on a linear scale for the temperature. In fig. 4a, we display the entropy as a function of temperature assuming fixedparameter J = 1 and ∆ = − . , for a range of J = { . , . , . , . , . } ,using conveniently a logarithmic scale to show the low-temperature behavior,close to the phase transition illustrated in fig. 2b. Where we show the influenceof zero temperature phase transition in the low-temperature limit, for | J | > . the system is highly influence by PAF(AFM) ground-state energy with residualentropy S → when T → , whereas frustrated energy contributes as low-lyingexcited energy, we can observe clearly residual entropy at J = 0 . is given by S → k B ln(3) when T → . However for J ? . , the system is dominated by afrustrated phase with residual entropy S → k B ln(2) when T → .In fig.4b, we also discuss another interesting thermodynamic quantity calledspecific heat ( C = − T ∂ f /∂T ). We illustrate for the same set of parameters13hose considered in fig.4a and also on a logarithmic scale. Here we observe ananomalous double peak in the low-temperature region, which was influenced bythe zero temperature phase transition between P AF ↔ F RU and the low-lyingenergy responsible for FM state[22]. For J = 0 . denoted by a solid (red)line, whose ground-state energy is non-degenerate and the first excited energyis macroscopically degenerate with a residual entropy S → k B ln(2) , when thetemperature increases the contribution of the degenerate energy becomes morerelevant than the contribution of the non-degenerate ground state, so the en-tropy curve is driven to behave like a frustrated system changing its concavityat T ≈ . , whereas for specific heat it manifests as a peak. Increasing thetemperature slightly more we observe another change of concavity in entropy at T ≈ . , this is because there is another phase transition in the neighborhood(PAF between FM), the contribution of the excited low-lying energy level againdrives the entropy (leading to a second peak in specific heat). A similar behaviorwas observed for a dashed (green) line assuming fixed J = 0 . with a changeof concavity in entropy at T ≈ . and a second change of concavity occurs at T ≈ . , thus in specific heat we observe a double peak. Whereas for J = 0 . (solid line), the change of the concavity in entropy at the lower temperature T ≈ . is almost imperceptible, that is because the ground sate energy ismacroscopically degenerate ( S → k B ln(2) ) and the lowest excited energy alsobecomes macroscopically degenerate ( S → k B ln(3) ), this is manifest as a smallpeak in the specific heat for T ≈ . . While the second peak in specificheat occurs at T ≈ . basically by the same mechanism of low-lying excitedenergies. An analogous behavior we observe for J = 0 . (magenta dashedline). Finally, for J = 0 . (doted-dashed line) where the ground-state energyis macroscopically degenerate ( S → k B ln(3) ), and the low-lying excited energy(that originates from FM ground-state energy) generating just one change ofconcavity at around T ≈ . .In fig. 4c, we plot the internal energy U = f + T S , for the same set ofparameters considered in fig. 4a, but here, we use a linear scale just to show thelow-temperature internal energy behavior, to relate with specific heat anomalouspeaks, since C = ∂U/∂T relates both quantities.
5. Conclusion
In this work we have proposed the Cairo pentagonal chain, motivated by re-cent discoveries of some compound such as the Fe lattice in the Bi Fe O andiron-based oxyfluoride Bi Fe O F compounds with a Cairo pentagonal tiling.Therefore, we proposed one stripe of the Cairo pentagonal Ising-Heisenberg lat-tice. Subsequently, we have discussed the phase transition at zero temperature,illustrating five phases: one ferromagnetic (FM) phase, one dimer antiferromag-netic (DAF), one plaquette antiferromagnetic (PAF), one typical antiferromag-netic (AFM) phase and one peculiarly frustrated (FRU) phase, where coexisttwo type of frustrated states with same energy but without mixing these phases,this kind of frustration is very unusual. It is worth to mention also, for the caseof two-dimensional pentagonal lattice the DAF phase, will be transformed into14 ferrimagnetic phase, due to the sharing spins between the unit cells. However,the AFM and PAF phase will be forbidden in a two-dimensional lattice, becausethe sharing spins between top and bottom unit cells will not be compatible.To study the thermodynamics of this model we have used the transfer matrixapproach and following the eight vertex model notation to find the partitionfunction. Using this result, we have discussed the entropy and specific heat as adependence of temperature. Accordingly, we observe an unusual behavior in thelow-temperature limit, such as residual entropy and the anomalous double peakdue the existence of three phases transition occurring in a very close region toeach other and one of them is frustrated state. Thus, the thermal excitation oflow-lying energy causes this anomalous double peak, and we also discussed theinternal energy in the low-temperature limit, where occurred this double peak. Acknowledgment
F. C. R. thanks Brazilian agency CAPES for full financial support. S. M. S.and O. R. thank Brazilian agencies, CNPq, FAPEMIG and CAPES for partialfinancial support. O. R. also thanks ICTP for partial financial support and thehospitality at ICTP.
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