Planckian effects on Ising model in an external electric field

In this article, we study Planckian effects on statistical thermodynamic properties of one dimensional polarized Ising model with an external electric field. We introduce Planckian effects by rewriting the Lagrangian of a static electromagnetic field in terms of deformed operators to obtain the corrected Lagrangian for a static electric field, and then deduce the corrected static electric field. Then we write down the corrected Ising model Hamiltonian in terms of the corrected static electric field. Corrected quantities such as the mean energy, Helmholtz free energy, and entropy are derived using the corrected Ising model Hamiltonian. Also, corrected quantities such as the polarization, dielectric constant, and polarizability tensor are derived. The expressions for the upper bounds of the deformation parameter α and the minimal length are written in terms of the dielectric constant. Using some experimental inputs we obtain the upper bound on the minimal length.


Introduction
It is well known that quantum gravitational effects should become significant at very short distances or very high energies, so the description of nature by using current theories such as quantum mechanics and general relativity is not (correct) complete.The effort for obtaining physical descriptions of very short-distance or very-high energy phenomena has led to the appearance of numerous theoretical models which are currently difficult to test.Surprisingly, most of these models predict the existence of an observable minimal length which was previously conceived by Heisenberg.[1][2][3] And the consideration of minimal length requires the modification of the usual Heisenberg uncertainty.The modified Heisenberg's uncertainty principle also known as generalized or gravitational uncertainty principle (GUP) [4][5][6][7][8][9][10] can be expressed as follows where p L is just the Planck length, and  is a dimensionless parameter.From Eq. (1), it can be seen that the limit 0   leads to the standard Heisenberg's uncertainty principle (HUP), and in the high- energy limit, Eq. (1) leads to a minimal distance which depends on the Planck length p L and the parameter  , () MIN P XL   .Some authors showed that the implementation of minimal length can be an effective way to deal with some infinities in quantum field theories.[11][12][13] Kempf and his collaborators developed an algebra which indicates that the minimal length can easily be deduced from the Heisenberg algebra in the deformed space.[14][15][16] In the Kempf algebra the relation between deformed operators is given by the following commutation relations ( ,0 Here , 1, 2,..., i j D  and ,'  are positive deformation parameters.The deformed operators i X , and i P are for position and momentum respectively.The combination of Eq. ( 2) with the Schwartz identity gives [17] 22 2 and this leads to the following expression for a minimal length [14]   Due to the importance of the minimal length, there have been various studies in the framework of GUP [17][18][19][20][21][22].The implementation of the minimal length leads to a modified Hamiltonian, so the usage of the modified Hamiltonian is enough to investigate properties of a system in the presence of minimal length.To the first order in the deformed parameters , and ' the deformed operators have the form [23] 22 where i x , and i i p ih  are the usual operators for position and momentum respectively.The same types of deformed operators commute when '2   , thus Eq. ( 2) reduces to: ,0 ,0 It was shown by Brau [24] that Eq. ( 6) is satisfied when the deformed position and momentum operators have the following form: The organization of the article is as follows: in Sec. 2 we modify the usual Lagrangian for an electrostatic potential to obtain the corrected electric field with a minimal length.We employ the corrected electric field to obtain the corrected Hamiltonian for one-dimensional polarized Ising model and the corrected Ising Hamiltonian is used to obtain corrected statistical thermodynamic properties and upper bounds.

Corrected electric field due to quantum gravitational effects
The Lagrangian for a static electric potential () x  with a charge density () x  can be expressed as follows where 0 is the permittivity of free space.
To obtain the corrected Lagrangian for the static electric field we just apply Eq. ( 7), which can be presented as follows where 2 ii    is the Laplace operator.In terms of operators from Eq. ( 9) the corrected Lagrangian has the following form After neglecting terms of order  2 and higher terms in Eq. ( 10) the corrected Lagrangian for a static electric field is written as follows Now, if we compare Eq. ( 11) with Eq. ( 8), we realize that the corrected electrostatic field has the following form Since the first term in Eq. ( 12) is just the usual electric field then we can conclude that the minimal length effect on the electrostatic field is ) where J is the dipole coupling constant between , ij which are electric dipole moments.The symbol , ij means that the sum over the sites , ij is restricted to nearest neighboring electric dipole moments.
With the periodic boundary conditions, the modified energy is as follows The associated modified partition function has the form The sum over , 1,..., i iN   can be computed using the method of transfer matrix.Using the following expression The modified partition function (15) can be expressed as follows ( , ) The modified partition function becomes (1 ) ( tr cosh (1 ) sinh ( 1) The modified partition function can be written using Eq. ( 20) in the following way From Eq. ( 20), it can easily be seen that GUP GUP    then Eq. ( 21) can be written as follows Since /1 GUP GUP    , in the thermodynamic limit N , the second term in ( 22) is negligible, and the modified partition function becomes . Using Eq. ( 22) the expression for the modified partition function becomes cosh (1 ) sinh ( 1)

Corrected statistical thermodynamic properties
The corrected partition function allows us to obtain modified thermodynamic properties.The corrected mean free energy is defined by If we substitute Eq. ( 23) into Eq.( 24) we find where The definition of the corrected Helmholtz free energy is If we substitute Eq. ( 23) into Eq.( 26) we find that The corrected entropy can be obtained either from Eq. ( 26) or the combination of Eqs. ( 25) and ( 27).
Using the combination, we obtain The usual statistical thermodynamic properties are recovered in the limit 0   .

Corrected electric properties
The corrected polarization per particle is defined by If we substitute Eq. ( 27) into Eq.( 29) we find that The corrected electric displacement is given by Since at very high energy (short-distance) the temperature is very high [25][26], then in the highenergy limit   0   , the Taylor expansion of Eq. ( 31) can be expressed as 1 ( 1) In the high-energy limit, the corrected dielectric constant is If we assume that the first term   0   as the standard constant, and the remaining expression as the term due to quantum gravitational effects, then we realize that we can write the following expression involving the dielectric constant  Since the minimal length scale is thought to be smaller than the Bohr scale, it can be seen that the correction to the dielectric constant would be easily observable if the minimal length is not far from the Bohr scale.If the minimal length is at the Planck scale, only a very sensitive technology would be able to measure the correction to the dielectric constant.

Summary
Our study consisted of studying Planckian effects on Ising model in an external electric field.The implementation of the minimal length showed that there is a correction to the usual electric field, so we utilized the corrected electric field to obtain the corrected Ising Hamiltonian.Then using this corrected Ising Hamiltonian we derived the corrected properties such as the mean energy, Helmholtz free energy, and entropy.We also derived the corrected polarization and corrected dielectric constant.
All usual properties are recovered when the deformation parameter  tends to zero.We expressed the upper bounds on  and  

3 .
One dimensional Ising model with a corrected electric field Consider one dimensional Ising model of N electric dipole moments k  in an external corrected electric field () GUP Ex .The corrected Hamiltonian for the polarized Ising model is written in terms of the corrected electric field.Thus, using Eq.(12), the corrected Hamiltonian for one dimensional Ising model is as follows 2 ,1

2 2
in terms of the relative modification of dielectric constant.Using some inputs we found the value of upper bound on   MIN x  .
The expression for the upper bound on the minimal length is 8