Abstract
We hereby introduce a research about a grand canonical ensemble for the extended twosite Hubbard model, that is, we consider the intersite interaction term in addition to those of the simple Hubbard model. To calculate the thermodynamical parameters, we utilize the nonextensive statistical mechanics; specifically, we perform the simulations of magnetic internal energy, specific heat, susceptibility, and thermal mean value of the particle number operator. We found out that the addition of the intersite interaction term provokes a shifting in all the simulated curves. Furthermore, for some values of the onsite Coulombian potential, we realize that, near absolute zero, the consideration of a chemical potential varying with temperature causes a nonzero entropy.
PACS
75.10.Jm, 05.30.d, 65.80.+n
Keywords:
Extended Hubbard model; Quantum statistical mechanics; Thermal properties of small particlesIntroduction
Currently, several researches exist on the subject of the application of a generalized statistics for magnetic systems in the literature [13]. Specially, we are encouraged by the recent interesting results for small magnetic systems obtained in [46]. Nevertheless, herein, we will utilize a different system to that utilized in the previously cited references. Thus, the scope of our investigation is the computer simulation on the onedimensional extended Hubbard model for M dimers by considering a grand canonical ensemble. The tool we utilize to calculate several thermodynamical parameters is the nonextensive statistical mechanics; along with it, we use the NewtonRaphson method for numerical approximations. With regard to applications, we must mention that, in the scientific literature, the organic compound called tetracyanoquinodimethane has been studied as a dimer gas [7,8]; also, there are studies on a dimerized Hubbard chain [9,10]. We expect our results to contribute to the clarification of the possible use of nonextensive statistical mechanics to research lowdimensional systems. Also, as a particular case of our outcomes, we expect to confirm previous results from the simple Hubbard model.
The Hubbard model was proposed in the early 1960s by the British physicist John Hubbard [1113]; basically, this model is the simplest one that takes into account the degrees of freedom linked to the electronic translational components. It has been applied to explain certain physical phenomena such as the metalinsulator transition, Mott insulators, ultracold atoms trapped in optical lattices, etc. [14,15]. On another side, in considering the several generalized statistical theories, the nonextensive statistical mechanics, also known as Tsallis statistics, is undoubtedly the most widely researched [1620]. It was invented by the Brazilian professor C. Tsallis as a theory that generalizes the BoltzmannGibbsShannon statistics [21]. Although several versions of the Tsallis statistics exist, in this article we will deploy the third version that was proposed in 1998 [22]. All of those versions differ in the way of defining the thermal mean values.
This paper is structured as follows: the section ‘Theoretical frame’ contains the theoretical aspects, the subsection ‘Twosite Hubbard model’ tackles the twosite Hubbard model, and the subsection ‘Nonextensive statistical mechanics’ deals with the nonextensive statistical mechanics. The section ‘Computer simulations’ introduces the utilized numerical procedure as well as the results of the computer simulations carried out. In the section ‘Conclusions’, we express the conclusions concerned with this investigation. Also, in the ‘ Acknowledgements ’, we thank the esteemed colleagues who provided useful information for this work.
Theoretical frame
In this section, we will display the fundamentals of the onedimensional extended Hubbard model as well as the elementary features of the nonextensive statistical mechanics. We will study the Hubbard model in a HilbertFock quantum space; also, we will show how to get the Tsallis distribution through the maximum entropy method.
Twosite Hubbard model
In a grand canonical ensemble, the one having a variable particle number, the Hamiltonian operator of the simple Hubbard model for a dimerized system is as follows:
where the indexes σ represent spins which may be up (↑) or down (↓), the indexes j designate sites 1 and 2 of the respective dimer, and t is the hopping integral for the kinetic energy term (the first one). Besides, in the framework of the second quantization, is the creation operator that originates a particle with spin σ in site 1, and c_{2,σ} symbolizes the annihilation operator that destructs a particle with spin σ in site 2. For the onsite interaction term (the second one), U stands for the Coulombian potential energy, n_{1,↑} represents the operator of particle number with spins ↑ in site 1 and n_{1,↓} symbolizes the operator of particle number with spins ↓ in site 1; all terms are similar for site 2. Lastly, in the third summand, h is an external magnetic field.
To take into account the extended Hubbard model [23],we need to add another energy term to Equation 1, namely the intersite Coulombian interaction:
with J_{1} and J_{2} denoting interactions between neighboring sites 1 and 2 inside each dimer; they are Coulombian repulsions modified by polaron effects. Consequently, we can group the two earlier equations to form the total Hamiltonian operator:
Energy eigenvalues and eigenvectors in the twosite Hubbard model
To attain the energy eigenvalues, we have to build the Hamiltonian matrix; for that purpose, in the context of the Dirac algebra, we make use of the following basis of 16 vectors:
Inside each ket, the comma symbol separates site 1 from site 2, and the spins may be up or down. Then, each matrix element of is obtained from the next bracketing operation:
that is, to get each of the 256 matrix elements, we must consider two steps: (1) apply on the respective ket and (2) apply the respective bra to the expression obtained from step 1. Making this, we have for step 1 (we lay down )
where we set x=U+J_{1}+J_{2}+h and y=U+J_{1}+J_{2}h. By applying step 2, we achieve the 16×16 Hermitian matrix of Equation 3:
where the symbol means represented by. After diagonalizing this mathematical object, we accomplish the energy eigenvalues given by the following:
with
Concerning the corresponding eigenvectors, they are the following:
the meanings of a_{1} and a_{2} are as follows:
Finally, we take advantage of the energy eigenstates to affirm they are also eigenstates of the magnetic dipolar momentum operator ; the respective kth eigenvalue of this last operator is as follows:
where n_{u} means the particle number with spins ↑ at the kth eigenstate, and n_{d} means the particle number with spins ↓ at the kth eigenstate. We want to emphasize these particle numbers are not evaluated at the states of Equation 4, but they are determinate from the states of Equation 10. Explicitly, the eigenvalues of are the following:
Nonextensive statistical mechanics
The Tsallis entropy underlies this theory which was postulated in 1988 [21]. The entropic form is shown as follows:
with p_{i} being the probability distribution to find the system in the ith state, represents p_{i} powered to the entropic index q, k_{B} is the Boltzmann constant, and symbolizes the quantum operation of trace over all states of the matrix . In Equation 14, the limit q tending to 1 allows us to recover the wellknown BoltzmannGibbsShannon entropy:
The nonextensive probability distribution p_{i} is obtained by application of the maximum entropy method, a procedure formulated by the American Edward T. Jaynes [24,25]. In that method we consider these constraints for a grand canonical system:
where E_{q} is the internal energy, ϵ_{i} denotes the energy eigenvalues of the energy operator , N_{q} is the quantum mean value of the particle number operator , and n_{i} is the eigenvalue of this last operator. As a final result of applying the maximum entropy method, we obtain the probability distribution
Z_{q} being the partition function:
where , β and μ are two out of the three Lagrange parameters  because of the three above constraints  utilized to optimize the Tsallis entropy, and the third parameter is one. Logically, we recover the standard distribution for q=1:
When we carry out the computer simulations, we will deploy the next definition of temperature T:
however, we must mention that apart of this definition, in the literature, there are other ones because so far the matter regarding temperature is an open problem [26]. In addition, in Equations 17 and 18, it is mandatory to take into account the Tsallis cutoff (to guarantee the positivity of the probabilities):
Thereupon, the distribution of probability can be compacted as follows:
Thus, we can state a critical value of temperature T_{c}. Above it, the ith probability is different from zero; below it, the ith probability is zero (but the other 15 probabilities which may be nonzero for T≤T_{c} exist). Considering k_{B}=1, this decisive temperature is as follows:
where E_{c} and N_{c} are respectively E_{q} and N_{q} evaluated at T=T_{c}. It is obvious that the last equation is a recursive formula. Thus, in the section ‘Computer simulations’, we will use the NewtonRaphson method to calculate the thermal mean values. On another side, the expressions for S_{q} and p_{i} can be rewritten utilizing the qexponential and qlogarithmic functions
thus, the replacement of these expressions into S_{q} and p_{i}, Equations 14 and 17, gives the following:
which, clearly, remember the entropy and the probability distribution for the BoltzmannGibbsShannon statistics. The quantum mean values of any observable, represented by the operator , in the HilbertFock space are calculated through the following formula:
where O_{i} stands for the ith eigenvalue of the observable . Naturally, the limit q→1 of the last expression becomes the known one
Thus, using Equations 17, 18 and 26, we have the internal energy and mean value of the particle number operator which can be expressed explicitly as follows:
Undeniably, these two thermodynamical parameters are defining two recurrence relations. These parameters will be found via NewtonRaphson method, as shown in the next section. Also, with respect to the magnetization, it is determined from the following formula:
where m_{i} stands for eigenvalues of the magnetic dipolar momentum operator . Additionally, other two thermodynamical parameters can be obtained if we derive the internal energy and magnetization, respectively. Thus, we obtain the specific heat
and the magnetic susceptibility
Finalizing this section, we want to lay emphasis upon the fact that it is the chemical potential μ that controls the results for the grand canonical ensemble. Thus, we will utilize the following relation to that parameter:
where μ_{0} represents the initial chemical potential, α is a constant, T is the changing temperature and T_{0} is the initial temperature.
Computer simulations
In this section, we display the numerical procedure we utilized and the computer simulations obtained from it.
Numerical procedure
As mentioned before, the formulas of Equation 28 are recursive, and it will be necessary to apply the NewtonRaphson method to find E_{q} and N_{q}. In consequence, we have to form two functions in which E_{q} and N_{q} will be the respective roots. Therefore, we define F_{1}(E_{q}, N_{q})≡F_{1}=0 with
and F_{2}(E_{q}, N_{q})≡F_{2}=0 with
Furthermore, the earlier mentioned method provides us the following iterative relations:
and
We will deploy as initial guesses those from the standard statistics, namely E_{q} and N_{q} with q=1.
Magnetic thermodynamical parameters
We will display computer simulations of the following magnetic thermodynamical properties: entropy per dimer, internal energy per dimer, specific heat per dimer, susceptibility per dimer, and mean value per dimer of the particle number operator. Furthermore, in order to reduce the parameters involved in the simulations, we will assume the relation J_{1}=J_{2}≡J for the intersite interaction term as well as k_{B}=1 for the Boltzmann constant. Even more, we define the normalized variables , , , and . Figure 1 shows the entropy vs. the normalized temperature, i.e., S_{q} vs. T_{t}, with entropic index values q= 1.0, 1.2, 1.4, 1.7 and 2.0. On the left side of Figure 1, we do not consider the intersite interaction, that is, J_{t}=0; we have U_{t}=1, 6 and 10, respectively, to Figure 1a, b, c. For these three graphics, we deploy h_{t}= 0 as well. In Figure 1a, at low temperature when T→0, S_{q}→0; the existence of a region where augmenting q means increasing S_{q} is apparent. However, at high temperature, we perceive that the landscape is completely opposite: augmenting q means decreasing S_{q}. With regard to Figure 1b, c, different nonzero values for entropy approaching to 0 exist; however, if we consider (independent of temperature), the entropy will be zero, as reported in [4]. On the right side of Figure 1, we take into account the intersite interaction term, that is, U_{t}= 1 and J_{t}= 0.2 in Figure 1d, U_{t}= 6 and J_{t}= 3 in Figure 1e, and U_{t}= 10 and J_{t}= 5 in Figure 1f; also, we lay down h_{t}= 0 to these three subfigures. In Figure 1d, it is evident that there is a small displacement towards the right side. However, this shift is enough notorious in Figure 1e, f. Likewise, in these last two graphics, we perceive that the entropy saturates so much before T_{t} borders on the absolute zero.
Figure 1. Entropy vs. normalized temperature. The values for q, U_{t}, J_{t} and h_{t} are indicated inside each graphic. On the left side (a to c), we have no intersite interplay, but on the right side (d to f), we take into account it.
In Figure 2, we exhibit the normalized internal energy vs. the normalized temperature, E_{q} vs. T_{t}; the values of q are 1.0, 1.2, 1.4, 1.7 and 2.0. On the left side of Figure 2, we have no intersite interaction, J_{t}= 0. We take into account U_{t}= 1 in Figure 2a, U_{t}= 6 in Figure 2b and U_{t}= 10 in Figure 2c; besides, we consider h_{t}= 0 in the three left subfigures. When contrasting Figure 2a from Figure 2b, c, we notice that the increase of U_{t} causes the expansion of the curves E_{q}. On the right side of Figure 2, we consider the interplay between neighboring sites for each dimer. Thus, we set J_{t}= 0.2, 3 and 5, respectively, to Figure 2d, e, f; the respective values of U_{t} and h_{t} are those from the left graphics. In Figure 2d, we realize the groundstate energy augments approximately to 1.4 (when it is contrasted with Figure 2a). However, in Figure 2e, f, we perceive the following: (1) the groundstate energy holds immutable when we add the intersite interaction term, and (2) in the low temperature range, the curves shift to the right side. However, at high temperatures, an increase of the values of E_{q} is visible.
Figure 2. Internal energy vs. normalized temperature. Inside each graphic, the respective values for q, U_{t}, J_{t}, and h_{t} are shown. Side by side (a to f), we have internal energy with and without intersite interaction.
Figure 3 displays graphs of the specific heat vs. the normalized temperature, C vs. T_{t}, with entropic index q= 1.0,1.2, 1.4, 1.7 and 2.0. In Figure 3a, we have U_{t}= 1, in Figure 3b we set U_{t}= 6 and in Figure 3c, we lay down U_{t}= 10. For all three earlier cases, J_{t}= 0 and h_{t}= 0. In Figure 3a, we have only a peak due to the antiparallel order; we perceive three regions verifying only one of the following statements: (1) the greater q, the lesser C and (2) the greater q, the greater C. In Figure 3b, c, we have two peaks: the first one is due to the antiparallel order, and the second one is provoked by the metalinsulator transition. In Figure 3b, three regions similar to those from Figure 3a are evident; however, between T_{t}≈ 0.7 and T_{t}≈ 1.7, the existence of a region that does not verify the above statements 1 and 2 is visible. In Figure 3c, four regions verifying either statement 1 or 2 exist. On another side, in contrast with the left side graphics, the right side graphics set the intersite interaction. Thus, we have J_{t}= 0.2, 3 and 5, respectively, in Figure 3d, e, f. In Figure 3d, an almost imperceptible movement of the curves towards the right side has happened. However, that displacement is more than noticeable in Figure 3e and Figure 3f. The intersite interaction causes the destruction of the second peak appearing in Figure 3b, as seen in Figure 3e. Notwithstanding, in Figure 3f, for q= 1, the peak due to antiparallel order maintains yet; for q≠ 1, that first peak does not exist anymore.
Figure 3. Specific heat vs. normalized temperature. The values for q, U_{t}, J_{t} and h_{t} are shown inside each graphic. The left side graphics (a to c) have no intersite interplay, but the right side graphics (d to f) present it.
Figure 4 brings forward the magnetic susceptibility vs. the normalized temperature, χvs. T_{t}, for q= 1.0, 1.2, 1.4, 1.7 and 2.0. In Figure 4a, b, c, we calculate χ by using respectively U_{t}= 1, U_{t}= 6, and U_{t}= 10; furthermore, we consider J_{t}= 0 and h_{t}= 0 for all three cases. In these curves without intersite interaction, we also find three regions verifying only one of the following affirmations: (1) the greater q, the greater χ and (2) the greater q, the lesser χ. Also, it is apparent that augmenting U_{t} means increasing the value of χ, a signal that the system is more localized. On another side, when we take into account the intersite interaction term J_{t}= 0.2 in Figure 4d, we detect a slight diminution of χ; however, the three regions from Figure 4a still exist. Nonetheless, augmenting the values of J_{t} provokes drastic drops for χ, as displayed in Figure 4e, f. Furthermore, we noticed that the addition of the intersite term caused the displacement of the curves towards the right side.
Figure 4. Magnetic susceptibility vs. normalized temperature. The values for q, U_{t}, J_{t} and h_{t} are indicated inside each graphic. On the left side (a to c), we have no intersite interplay. However, on the right side (d to f), we have that interaction.
Lastly, in Figure 5, the thermal mean value of the particle number operator vs. the normalized temperature, i.e., N_{q} vs. T_{t}, with q= 1.0, 1.2, 1.4, 1.7 and 2.0, is presented. On the left side of Figure 5, we have U_{t}= 1, 6 and 10, respectively, in Figure 5a, b, c. For all of them, J_{t}= 0 and h_{t}= 0. In Figure 5a, we noticed that when T_{t} is zero, N_{q} saturates at 2; in increasing T_{t}, N_{q} drops but it augments again and tends towards 2 in high temperatures. The case of Figure 5b, c is completely different: around T_{t}= 0, N_{q} saturates at 1, and it never decreases. On the right side of Figure 5, we consider the intersite interaction. In Figure 5d, we see that this term causes N_{q} to lower slightly its value as T_{t} approaches 0.5. However, in Figure 5e, f, it was detected that the intersite interaction originates a drop that does not exist neither in Figure 5b nor in Figure 5c.
Figure 5. Mean value of particle number operator vs. normalized temperature. Inside each graphic, the respective values for q, U_{t}, J_{t} and h_{t} are indicated. On the left side (a to c), we have no intersite interplay, though on the right side (d to f), we have that interaction.
Conclusions
In this article, we have introduced a research to calculate thermodynamical properties from a grand canonical ensemble of the extended twosite Hubbard model. The Tsallis statistics was utilized instead of the standard one because it would be more appropriate to study lowdimensional systems for it exists several investigations in that regard. As concluding remarks, we can affirm that we have verified early results for the simple twosite Hubbard model. Also, we have found out that the addition of the intersite interaction term to the simple Hubbard model provoked a displacement of the curves of entropy, internal energy, specific heat, susceptibility, and mean value of the particle number operator, respectively; additionally, we have perceived that, near absolute zero, the consideration of a chemical potential varying with temperature causes a remnant of entropy, but it happens only at some values of the onsite Coulombian potential. A way of understanding the displacement of the curves is realizing the critical temperature changes because the Tsallis cutoff is satisfied with new conditions, i.e., the intersite interaction term determines new values for critical temperatures.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
FARN carried out the outline of the paper as well as proposed the main formulas. ECTT participated by providing additional formulas. PPP participated in critiquing and correcting the original draft. All authors read and approved the final manuscript.
Authors’ information
FARN is a Peruvian and a doctor in physical sciences at the Brazilian Center for Physics Research (CBPF) located in Rio de Janeiro, Brazil. ECTT is a Peruvian and a doctor in physics at the Universidade Federal do Rio de Janeiro (UFRJ) located in Rio de Janeiro, Brazil. PPP is a Peruvian and a licentiate in physics at the Universidad Nacional Mayor de San Marcos (UNMSM) located in Lima, Peru.
Acknowledgements
We are very thankful to Professor M. Matlak  from the University of Silesia, Poland  for giving us updated references to elaborate this paper.
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