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We study the correlation functions of one-dimensional Hubbard model in the presence of external magnetic field through the conformal field method. The long distance behaviour of the correlation functions and their unusual exponents for the model in the presence of a magnetic field are developed by solving the dressed charge matrix equations and setting the number of occupancies
to one, as alternative to the usual zero used by authors in literatures. This work shows that the exponent of the correlation functions is a monotonous function of magnetic field and the correlation functions decay as powers of these unusual exponents. As the magnetic field goes to zero, we obtain the exponents as 8.125, 11.125, 17.125, 26.125 and 38.125 at k_{F}, 3k_{F}, 5k_{F}, 7k_{F}
and 9k_{F}. Our analytical results will provide insights into criticality in condensed matter physics.

Almost twenty five years ago, Frahm and Korepin introduced the calculation of critical exponents for the one-dimensional (1D) Hubbard model, using the finite size scaling and the principle of conformal field theory (CFT) [

(with correlation exponents

8.125, 11.125, 17.125, 26.125 and 38.125 around

nent of the correlation function changes monotonically with change in magnetic field. The progress made in the understanding of critical phenomena in quantum systems as a result of conformal invariance have provided great insights to the problem of calculation of these critical exponents [

field correlation function and the density-density correlation function by setting the parameter

investigate how this affects the conformal dimensions and critical exponents of the correlation functions. This paper is organized as follows. In Section 2, we review the Bethe Ansatz equations of the Hubbard model and the analytic form of the correlation functions predicted by CFT is given. The dressed charge matrix elements are also calculated with the Wiener-Hopf technique and these elements are used to obtain the magnetic field dependence of the conformal dimensions. The long-distance behaviour of the electron field and density-density correlation functions and their unusual exponents for small magnetic field are calculated in Section 3. The electron field correlation function in momentum space and their Tomonaga-Luttinger (TL) liquid behaviour is examined in Section 4. Finally, Section 5 is devoted to discussion of the properties of the critical exponents for

The Hubbard model is basically the simplest model describing interacting spin-1/2 fermions in many-body physics. In the presence of magnetic field it is defined by the Hamiltonian [

where

number operator. u is the on-site Coulomb repulsion, μ is the chemical potential and H is the external magnetic field. The hopping integral t = 1. Lieb and Wu [

where the quantum number I_{j} and J_{α} are integers or half-odd integer, _{↑} and N_{↓} being the number of electrons with spin up and down, and N_{s} = N_{↓} down spins are characterized by the moment a k_{j} of holons and rapidities λ_{α} of spinons.

In the thermodynamic limit, with continuous momentum and rapidity variables, the Lieb-Wu equations become integral equations for the ground state distribution functions of moment a

The state corresponding to the solution of Equations (2) and (3) has energy and momentum given by

where the conformal dimensions are given by

The positive integers

change in the number of electrons (down-spin) with respect to the ground state,

and the elements are defined by the solutions of the following coupled integral equations

where the kernel is defined as

The values of

For small magnetic field we solve the dressed charge matrix equations by Wiener-Hopf technique [

Fourier transforming Equation (17), we obtain

where the kernels are given by

We solve Equation (18) by introducing the function

and expanding it as

where

The driving terms

Where the functions

Also we assume

In terms of these functions we express the Fourier transform of Equation (23) as

where

where

Useful special function of

Using Equations (27) and (28), we obtain

Decompose the right hand side of Equation (32) into the sum of two functions

This implies that

To obtain the solution of Equation (22) for

We decompose the first term by using

The second term of Equation (35) is meromorphic function of

Note, there is no pole at

Using Equation (39) we can express the function

Applying the formula Equations (41) to (35) and Equation (33), we obtain

Now,

Therefore,

For

The functions

From Equation (23) for

By definition

where

As

Simplifying further, we obtain

Using Equation (51), we obtain

Next, the second order contribution to

From Equation (28)

From Equation (33),

We have decomposed

where

From Equation (29), as

Since,

For

From Equation (34), we obtain

Using

we obtain

Using the value of

Therefore, with Equations (55) and (69), we obtain

Now to evaluate the dressed charge matrix element

Applying the same process in the determination of Equation (70), we obtain

Similarly, with the same process, we obtain the other two elements of the dressed charge matrix as

and

From Equation (16) together with the property that

Using Equation (75) on Equations (70) and (72), we obtain the dressed charge matrix equations as

At half-filling

To obtain the conformal dimensions in terms of small magnetic field we use Equations (80) to (83) on Equations (7) and (8). Note that,

Therefore, the magnetic field dependence of the conformal dimensions are given by

According to the principles of CFT, the general expression for correlation function contains factors from holons and spinons, given by [

We now use the results obtained in the last section to obtain the magnetic field dependence of the unusual exponents of the electron field correlation function and density-density correlation function by setting the non-

negative integer

from the quantum numbers (D_{c}, D_{s}) = (1/2, −1/2), (3/2, −3/2), (5/2, −5/2), (7/2, −7/2), (9/2, −9/2), ΔN_{c} = 1 and ΔN_{s} = 0. Therefore, the corresponding conformal dimensions for

where the contributions from

The critical exponent is given by

This implies that

and

Next, we obtain the conformal dimensions for

Using Equations (97) and (99) on Equation (87), we obtain

The critical exponent is given by

and

Next, for

Using Equations (104) and (106) on Equation (87), we obtain

The critical exponent is given by

and

For

Using Equations (111) and (113) on Equation (87), we obtain

The critical exponent is given by

and

Finally, for

Using Equations (118) and (120) on Equation (87), we obtain

The critical exponent is given by

and

Combining Equations (92), (100), (107), (114) and (121), we obtain the long-distance asymptotic form of the electron field correlation function with up-spin as

Lastly, we consider the density-density correlation function which originates from the quantum numbers

Again contributions from

The critical exponents are given by

and

For

Using Equations (130) and (131) on Equation (87), we obtain

The critical exponents are given by

and

Next, for

Using Equations (135) and (136) on Equation (87), we obtain

and

Finally, for

Using Equations (140) and (141) on Equation (87), we obtain

and

The electron field correlation function Equation (124) has singularities at the Fermi points

The critical exponent

and

Here we neglect logarithmic field dependence. Equation (145) represents the momentum distribution function around

Another singularity is at

with critical exponent

Equation (148) exhibits a typical power-law singularity of the TL liquid around the Fermi point

Next at

with the unusual exponent

Also, Equation (150) represents the momentum distribution function around the Fermi point

At

and

Equation (152) exhibits a typical power-law behaviour of the TL liquid around

Finally, at

with

and Equation (154) also exhibits typical power-law behaviour of the TL liquid around the Fermi point

In this paper, we have calculated the electron field and density-density correlation functions and their unusual exponents by using the nonnegative integer

from the excitation of

In conclusion, the electron field correlation function and the unusual exponents has been obtained around the Fermi points