Tomonaga-Luttinger Unusual Exponents around Fermi Points in the One-Dimensional Hubbard Model

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 c s N , ± 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 F F F F k k k k , 3 ,5 ,7 and F k 9 . Our analytical results will provide insights into criticality in condensed matter physics.


Introduction
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) [1].This enabled theorists to explore the physics of 1D correlated electron systems.Notwithstanding significant works, the understanding of the behaviour of correlated electron systems is not yet complete.In one dimension, the Hubbard Hamiltonian provides opportunity to study correlation effects in 1D models and the cor-relation functions decay as power of the distance [2]- [4].It is the calculation of the critical exponents characterizing this power-law behaviour that have attracted constant theoretical interest.Outstanding results in this field (with correlation exponents .125,1.12 d 1 0 5 an v = for , 0 c s N ± = at zero magnetic field around the Fermi points ,3 and 5 k k k ) have been obtained from conformal field techniques, perturbation calculations and renormali- zation group methods in different models [1] [5]- [7].For our calculation, we obtain the correlation exponents as 8.125, 11.125, 17.125, 26.125 and 38.125 around 3 ,3 ,5 ,7 k k k k and 9 F k by setting the parameter cha- racterizing particle-hole excitation to one ( ) , 1 c s N ± = as the magnetic field goes to zero, and the unusual exponent 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 [8].Although, interacting 1D quantum systems might carry countless low-energy excitations, with linear dispersion relations, but with different Fermi velocities, so the systems are not Lorentz invariant [9].When the motions of these excitations are decoupled, one can now apply the CFT [10].Usually, in the application of the conformal field techniques, the nonnegative integer where is the creation (annihilation) operator with electron spin σ at site j and † , , , j j j n c c σ σ σ = is the 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 [2] has solved Equation (1) exactly and obtained the Bethe Ansatz equations where the quantum number I j and J α are integers or half-odd integer, c N N N with N ↑ 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 and the elements are defined by the solutions of the following coupled integral equations where the kernel is defined as The values of 0 0 and k λ are fixed by ( ) ( ) For small magnetic field we solve the dressed charge matrix equations by Wiener-Hopf technique [12] [13] for terms up to order 1 u in the strong coupling limit.With Equation ( 16), we write Equation (13) as 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 ( ) n y λ are defined as the solutions of the Wiener-Hopf equations The driving terms ( ) n g λ and the solutions ( ) n y λ becomes smaller as n increases because λ is large.Our procedure follows Fabian et al. [11].Assuming the function Also we assume In terms of these functions we express the Fourier transform of Equation (23) as where ( ) n g ω  is the Fourier transform of ( ) n g λ .Now we split Equation (23) into the sum of two parts that are analytical and non-zero in the upper and lower half planes.To obtain this we use the factorization ( ) ( ) ( ) ± are analytic and non-zero in the upper and lower half planes respectively and are normalized as 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 .
To obtain the solution of Equation ( 22) for ( ) 0 y λ , we set the driving term to be ( ) ( ) ( ) We decompose the first term by using ( ) ( ) The second term of Equation ( 35) is meromorphic function of ω with simple poles located at ( ) Note, there is no pole at 0 ω = .The decomposition of the factor ( ) Applying the formula Equations ( 41) to (35) and Equation (33), we obtain 2 cosh 2 .
The functions ( ) where H is magnetic field, c H is critical field, u strong coupling, 0 H magnetic field at zero temperature and 0 λ corresponds to Fermi points.Combining the result Equation (49) with Equation (50), we obtain the first order contribution to ss Z as follows As 0 ε → , we use Equations ( 30) and (31) on Equation (52) to obtain ( ) Simplifying further, we obtain ( ) ( ) Using Equation (51), we obtain From Equation (33), We have decomposed which is analytic in the upper and lower half-planes.
( ) where ε is a small positive constant.

G x
+ has a branch cut along the negative imaginary axis and by deforming the contour of integration we rewrite Equation (58) as From Equation (29), as ix ω → ( ) ( ) For 0 x > the integrand rapidly decrease because 0 1 λ  , and hence the integral is approximated by ex- panding the terms other than From Equation (34), we obtain ( ) ( ) (0) lim , and lim 1, we obtain ( ) Using the value of 0 λ from Equation (51) in Equation (68), we obtain Therefore, with Equations ( 55) and (69), we obtain Now to evaluate the dressed charge matrix element ( ) 0 , sc Z k we take the Fourier transform of Equations ( 16) and ( 18) and obtain 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 1 c n = , and by neglecting corrections to order ( ) 1 u , the elements of the dressed charge become To obtain the conformal dimensions in terms of small magnetic field we use Equations ( 80) to (83) on Equations (7) and (8) 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 [11] ( ) ( , exp 2 exp 2 , .
x iv t x iv t x iv t x iv t

Correlation Functions in Magnetic Field
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 nonnegative integer ΔN s = 0. Therefore, the corresponding conformal dimensions for ( ) ( ) where the contributions from ( ) ( ) Next, we obtain the conformal dimensions for ( ) ( ) Using Equations ( 97) and (99) on Equation (87), we obtain .
The critical exponent is given by , we obtain the conformal dimensions as ( ) Using Equations ( 104) and (106) on Equation (87), we obtain ( ) The critical exponent is given by Using Equations ( 111) and (113) on Equation ( The critical exponent is given by , we obtain the conformal dimensions as ( ) 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   125) and (126) on Equation (87), we obtain ( ) The critical exponents are given by x iv t x iv t The critical exponents are given by Using Equations ( 135) and (136) on Equation (87), we obtain ( )

Correlation Function in Momentum Space
The electron field correlation function Equation ( 124) has singularities at the Fermi points , , The critical exponent Here we neglect logarithmic field dependence.Equation (145) represents the momentum distribution function around F k for the electron field correlator.It exhibits a typical power-law behaviour of the TL liquid, with critical exponent given by Equation (146).This unusual exponent 8.125 ν → as the magnetic field goes to zero.
Another singularity is at ,  ν → as the magnetic field goes to zero.

Discussions
In this paper, we have calculated the electron field and density-density correlation functions and their unusual exponents by using the nonnegative integer N ± = in the strong-coupling limit has been obtained before [1] [3] [14]- [16].At zero magnetic field ν is a monotonous function of the coupling constant u.In our calculation, we have used N ± = indicates presence of particle-hole excitations in the asymptotics.Therefore, the results obtained here are contributions from both particle-hole excitations and collective modes (holons and spinons).

(
, c s N ± , for holon and spinon describes particle-hole excitations, with ( ) occupancies that a particle at the right (left) Fermi level jumps to, number of electrons (down-spin) with respect to the ground state, c D represents the number of particles which transfer from one Fermi level of the holon to the other and s D represents the number of par- ticles which transfer from one Fermi level of the spinon to the other, and both c D and s D are either integer or half-odd integer values.Finally, the dressed charge matrix Z describing anomalous behaviour of critical exponents is given by , the upper and lower planes respectively, with in the lower half-plane as the sum of two functions

=
55)Next, the second order contribution to ( ) ( ) is obtained by taking the Fourier transform of Equation (23) for 1 n = .

N
± characterizing particle-hole excitations as 1 in the 1D Hub- bard model.Based on the principles of CFT, we obtain expressions for the unusual exponents that describe the long-distance behaviour of the correlation functions in coordinate and momentum space.The unusual behaviour of the exponents depend on the magnetic field.The zero magnetic field 0 and obtain the exponents8.125,11.125, 17.125, 26.125 and 38.125   v respectively, and observe that ν is a monotonous function of the magnetic field.The F k part arises from the excitation of ( k and 9 F k oscillation parts respectively.In conclusion, the electron field correlation function and the unusual exponents has been obtained around the Fermi points .These results show that the correlation function clearly exhibits power-law behaviour of TL liquids as the exponent v changes monotonically with change in magnetic field . Note that, Equation (150)represents the momentum distribution function around the Fermi point 5 F k for the electron field correlator, and it exhibits a typical power-law behaviour of the TL liquid with critical exponent given by Equation (151).