Comparative Study of a Critical Behavior of a Coupled Spin-Electron Model on a Doubly Decorated Square Lattice in the Canonical and Grand-Canonical Ensemble
VVol. XXX (20XX)
CSMAG’19
No. X
Comparative Study of a Critical Behavior of a CoupledSpin-Electron Model on a Doubly Decorated Square Latticein the Canonical and Grand-Canonical Ensemble
H. ˇCenˇcarikov´a ∗ and N. Tomaˇsoviˇcov´a Institute of Experimental Physics, Slovak Academy of Sciences, Watsonova 47, 040 01 Koˇsice,SlovakiaThe critical behavior of a hybrid spin-electron model with localized Ising spins placed on nodalsites and mobile electrons delocalized over bonds between two nodal lattice sites is analyzed bythe use of a generalized decoration-iteration transformation. Our attention is primarily concen-trated on a rigorous analysis of a critical temperature in canonical and grand-canonical statisticalensemble at two particular electron concentrations, corresponding to a quarter ( ρ = 1) and a half( ρ = 2) filled case. It is found that the critical temperature of the investigated spin-electronsystem in the canonical and grand-canonical ensemble may be remarkably different and is verysensitive to the competition among the model parameters like the electron hopping amplitude( t ), the Ising coupling between the localized spins ( J (cid:48) ), the electrostatic potential ( V ) and theelectron concentration ( ρ ). In addition, it is detected that the increasing electrostatic poten-tial has a reduction effect upon the deviation between critical temperatures in both statisticalensembles. Keywords: spin-electron model, generalized decoration-iteration transformation, critical be-havior
1. Introduction
The study of the critical behavior is one of the most extensively studied problem in a condensedmatter physics, with an aim to deeper understand processes realized in the vicinity of a crit-ical point. The most effective treatment from the theoretical point of view is the applicationof traditional analytical approaches or state-of-the-art numerical methods on suitable lattice-statistical models. Based on the method as well as model complexity, the choice of the statistical ∗ Corresponding author: [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] A ug Template for Calculation of Length of Manuscript . . . ensemble may be crucial to resolve a defined mathematical problem. Although in most text-books the equivalence between different ensembles in the thermodynamic limit is illustrated viathe Van Hove theorem [1], there exists a few studies which indicate on the ensemble inequality(see Ref. [2] and Refs. 5-17 therein). In the present work, we will examine a relatively simpledoubly decorated spin-electron model on a square lattice (DDSEM), which was successfully usedto elucidate the origin of some unconventional phenomena in coupled spin-electron systems likea doping-induce crossover from ferro- to antiferromagnetism [3, 4], an enhanced magnetoelectriceffect [5], metamagnetic transitions [6] or magnetic re-entrance [7]. Our main attention will beconcentrated on the comparative study of a critical behavior in the canonical ensemble ( CE )and grand-canonical one ( GCE ), with a motivation to examine if the choice of the statisticalensemble could have a significant impact on a critical temperature in the DDSEM, where theintrinsic features can be modified by an extrinsic factor like an electric field.
2. Model and methods
Let us consider an interacting spin-electron model on the square lattice (see Fig. 1 in Ref. [8]),which consists of Ising spins localized at nodal lattice sites and mobile electrons delocalized overtwo decorating atoms between two nearest-neighbor Ising spins. Due to the local character ofall assumed interactions, the total Hamiltonian can be divided into the sum of 2N commutingbond Hamiltonians ˆ H = (cid:80) Nk =1 ˆ H k , whereˆ H k = − t (cid:88) γ = ↑ , ↓ (ˆ c † k ,γ ˆ c k ,γ + H.c. ) − J (cid:48) ˆ σ zk ˆ σ zk − (cid:88) α =1 , (cid:104) J ˆ σ zk α (ˆ n k α , ↑ − ˆ n k α , ↓ ) − ( − α V / √ n k α (cid:105) . (1)Here ˆ c † k α ,γ and ˆ c k α ,γ ( α = 1 , γ = ↑ , ↓ ) denote the usual creation and annihilation fermionicoperators, for which the respective number operators read ˆ n k α = (cid:80) γ ˆ n k α ,γ and ˆ n k = (cid:80) α ˆ n k α .In addition, ˆ σ zk α is the z component of the Pauli operator with the eigenvalues σ zk α = ±
1. TheHamiltonian (1) is formed by the kinetic energy of mobile electrons modulated by the electronhopping ( t ) and exchange interactions of Ising-type between (i) the nearest-neighbor Ising spins( J (cid:48) ) and (ii) the Ising spins and their nearest-neighbor mobile electrons ( J ). Finally, the Hamil-tonian also involves the electrostatic energy for a pair of mobile electrons modulated throughthe electrostatic potential V originating from the applied electric field acting along the crys- Template for Calculation of Length of Manuscript . . . tallographic axis [11] . This spatial field orientation results in the identical influence of electricdimers on horizontal and vertical bonds with a magnitude of
V / √
2. We recall, that for a correctanalysis of the critical behavior in the
GCE , the Hamiltonian (1) has to be extended with aterm − µ ˆ n k , involving the chemical potential µ , which controls the bond electron density n k .To study the critical behavior of the investigated model in both ensembles, we use the rigorousapproach based on the decoration-iteration transformation [9, 10, 11]. This procedure, describedin detail in Refs. [5, 7], results to the unique relation Ξ( T, J, J (cid:48) , t, V ) = A N Z IM ( T, R ) betweenthe grand-canonical or canonical partition function Ξ of the investigated model and the canonicalpartition function Z IM of a pure Ising model with new effective parameters A and R . Based ontheir knowledge we are able to determine the critical temperature T c as a solution of the criticalcondition sinh (2 β c R ) = 1, in which β c = 1 /k B T c and k B is a Boltzmann constant.
3. Results and discussion
Before a detailed discussion, we note that all analyzes are performed for a ferromagnetic (F)Ising interaction
J >
0, since the antiferromagnetic (AF) one leads to the preference of thephysically identical magnetic structures in which each Ising spin is replaced with its equivalentoriented antiparallelly to the electron spins. Two particular electron concentrations ρ = 1 and ρ = 2 are assumed, where ρ = (cid:80) Nk n k / N . The most interesting results are collected at Fig. 1,where the normalized critical temperature k B T c /J against the electron hopping t/J is plottedfor both electron concentrations. The detail inspection of the J (cid:48) /J = 0 case feels out an evidentensemble inequality between critical temperatures ∆ = k B ( T CEc − T GCEc ) /J for both investigated ρ . Obviously, the system with a fixed number of particles ( CE ) is a more resistant to thermalfluctuations as its counterpart with a variable number of particles ( GCE ), resulting to the equalor higher value of k B T c /J . The possible explanation of this interesting observation lies in the fact,that in the GCE , the system with a defined ρ is a mixture of totally 2N independent bonds withdifferent eigenvectors conditioned by various number of electrons ranging from n k = 0 to n k = 2( ρ = 1), or to n k = 4 ( ρ = 2). These different eigenvectors exhibit also a different magnetism, F orAF one [5] and thus, the system is slightly ’disordered’. Due to this fact, the magnetic ordering ofthe system in the GCE is destroyed easier than in the CE , where all 2N bonds of the investigated Template for Calculation of Length of Manuscript . . .
Fig. 1:
Finite temperature phase diagrams in the k B T c /J − t/J plane for several values of model param-eters. GCE (solid lines) and CE (dashed lines) results. Lower parts of each figures present a behaviorof ∆ against the t/J . model exhibit just the one type of magnetic ordering. In addition, it is clear from Fig. 1 that theensemble inequality ∆ strongly depends on the type of the ground-state arrangement. Whereasin the F phase the ∆ increases with increasing t/J up to its saturation value at t/J → ∞ , inthe AF one the ∆ reaches nonzero values exclusively at small or intermediate values of t/J . Wesuppose that the origin of this different behavior is attributable to a charge distribution over thebond upon the t/J modulation resulting to changes of an AF and non-magnetic (NM) characterof respective electron dimers with n k = 2. It is noteworthy to mention that such electron dimerscan be likewise found in a quarter filling if the GCE is taken into account. Moreover, it isdetected that the ensemble inequality ∆ can be easily reduced by switching on the
V /J , whichsimilarly as t/J favors charge segregation instead of the homogeneous electron distribution.The evident ensemble inequality in a critical temperature is also detected for a nonzerointeraction J (cid:48) /J , see Fig. 1. Moreover, in the regime where the | J (cid:48) /J | > ρ = 1 , J (cid:48) /J = 0 . ρ = 2 , J (cid:48) /J = − .
2) a direct coherence between Ising spins leads to theenhancement of the magnitude as well as the area of ∆. The qualitative character of all previousconclusions remains unchanged. The situation is slightly different in the regime, where thecompetition among all model parameters leads to the existence of quantum magnetic phase
Template for Calculation of Length of Manuscript . . . transitions driven by the electron hopping t/J . In this regime the ensemble inequality ∆ canreach besides positive values also negative ones exclusively detected at t/J →
0. The explanationof an enhancement of a critical temperature at ρ = 2 in the GCE against the CE may be foundin a fully F character of bonds with an odd number of n k rarely distributed over the remainingdouble occupied bonds with a parallel orientation of Ising spins accompanied by a superpositionof electrons in AF and NM states. On the other hand, an enhancement of a critical temperatureat ρ = 1 in the GCE against the CE has its origin in a presence of a few AF bonds with n k = 2,which dilute remaining bonds with n k = 1 arranged in novel AF formation [7]. The mixturedAF/NM character of the electron subsystem at AF bonds with n k = 2 has a strengtheningeffect on the stability of magnetic arrangement in contradiction to the F one observed at n k = 1.Similarly as in the J (cid:48) /J = 0 case, the nonzero electrostatic potential V /J suppresses the ensembleinequality in a critical temperature, however the reduction effect is slightly damped by nonzerointeraction between nearest-neighbor Ising spins, see Fig. 1.Another interesting observation which directly follows from the Fig. 1 is an absence of thethermally driven re-entrant magnetic phase transition in the CE , which is a very specific featuredetected on the DDSEM with a square plaquette in the GCE [7]. The representative situationis shown in Fig. 1 for ρ = 1, J (cid:48) /J = − . V /J = 0, however, an identical observationhas been detected in a whole parameter space, where the thermally driven re-entrant magnetic-phase transition has been detected assuming the
GCE . Based on this interesting observation, wesuppose that the thermally driven re-entrant magnetic-phase transition observed in real materialsis an intrinsic feature of materials and originates from the different number of valence electronsbetween two nearest-neighbor localized magnetic ions. Consequently, to bring a deeper insightinto the understanding of this unconventional phenomenon, the respective theoretical analyzesshould be performed using the
GCE .
4. Conclusions
We have performed a comparative rigorous study of a critical behavior of the DDSEM on asquare lattice in the CE and GCE . It has been found that there exist regions, where the criticaltemperature and thus the stability of magnetically ordered phases can be dramatically different
Template for Calculation of Length of Manuscript . . . in the system with a fixed ( CE ) or fluctuating ( GCE ) number of particles. As was discussed, thecrucial impact on the magnitude of ∆ lies in the number of doubly occupied bonds with a chargesegregation at one of two decorating sites, which can support or reduce the effect of thermalfluctuations on the stability of magnetic orderings. In addition, it was demonstrated that theintriguing thermally driven re-entrant magnetic phase transition observed in real materials canbe theoretically described only by the
GCE and is fully absent in the CE counterpart. Finally, itwas shown that the ensemble inequality ∆ is gradually reduced by applying the external electricfield along the [11] crystallographic axis.This work was supported under the grant Nos. APVV-16-0186, VEGA 1/0043/16 and ITMS26220120047. References [1] K. Huang,
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