Momentum Distribution Critical Exponents for 1 D Hubbard model in a Magnetic Field

Critical exponents at , 3 , 5 F F F k k k and 7 F k for the momentum distribution function are studied for one-dimensional Hubbard model in the presence of magnetic field, using conformal field theory (CFT) approach. Exponents at F k and 3 F k are reproduced. Results at 5 F k is in contrast to earlier numerical prediction of 1, while at 7 F k , the exponent is 49/8. The singularities at 5 F k and 7 F k appears to be weak and gradually degenerating into a smooth curve.


Introduction
Despite one-dimensional (1D) Hubbard model (HM) is the simplest strongly correlated model used to describe the physics of correlated electron systems, the understanding of this model is not complete.This is why calculation of asymptotic correlation functions, momentum distribution (MD) and its critical exponents have not been properly resolved around odd Fermi points.In particular, critical exponent at 3 F k obtained by Shaojin et al. [1, 2] disagrees with Ogata and Shiba [3, 4].
The ground state of the 1D HM is a Tomonaga-Luttinger (TL) liquid and the MD function does not show sharp jump, but rather a power-law singularity near the Fermi surface [5].Ogata and Shiba carried out numerical calculations for critical exponents of MD near F k and 3 F k , and obtained 1/8 and 9/8.Soon after, these critical exponents were reproduced analytically [6,7].Thereafter, Shaojin et al. [1,2] carried out Density Matrix Renormalization Group numerical calculations and found that a power-law singularity shows up at We have in this work extended the calculation of critical exponents which has only been done for This paper is outlined as follows.Section 1, illustrates the motivation to this study.The mathematical theory is reviewed in section 2. While in section 3, we present analytical calculations of the asymptotic correlation function, MD and critical exponents.Discussion of results and conclusion are shown in section 4 and this is immediately followed by list of references.

One-dimensional Hubbard Model and Conformal Field Theory approach for Correlation Functions
The Hamiltonian [10] of the HM is given by the expression Where   † ,, jj cc  is the creation (annihilation) operator with electron spin σ at site j and the number operator is † , , , . u is the on-site Coulomb repulsion, μ is the chemical potential and H is the external magnetic field.The hopping integral 1 t  .Lieb and Wu (1968) obtained the Bethe-Ansatz solution to Eqn. (2.1) as Where the conformal dimensions () cs   for the holon (spinon) excitations are given by The positive integers , cs N  , for holon and spinon describes particle-hole excitations, with Where the kernel is defined as 12) The values of (2.13) For small magnetic field Eqns.(2.8) -(2.11) has been solved by Wiener-Hopf technique [8,11] for terms up to order Where B is the external magnetic field, c B is the critical field and 0 B is the magnetic field at zero temperature and these are related by with u as the strong coupling.However, the Hubbard model is critical at zero temperature [7 -9] and remarkable progress in the description of critical phenomena has been made by application of the concept of CFT [7,13].In the language of conformal field theory, the two point correlation function [7,10] is given by

Discussion
It is clear from the critical exponent equations given by (3.16), (3.18), (3.20) and (3.22), that exponents of the momentum distribution which characterizes the Tomonaga-Luttinger liquid in contrast to the usual Fermi liquid theory depends on the magnetic field.When 0 B  , the critical exponents are obtained as 1/8, 9/8, 25/8 and 49/8 at various Fermi points.
Power-law singularity occurs at the Fermi points , 3 , 5 F k may lead to better understanding of Tomonaga-Luttinger liquid.This can be examined further to explore the physics involved.

3 F k with critical exponent 3 / 4 , and 1 at 5 Fk
. The result of Shaojin et al. disagrees with Ogata and others [3 -7] at 3 F k .

F k and 3 F 5 F k and 7 Fk 5 F k and 7 Fk 5 F k and 7 Fk
k with the CFT technique[7 -9], by obtaining MD and critical exponents at .In this study, we used the methods of CFT to establish power-law dependence of the MD and obtained critical exponents at these new Fermi points.The new critical exponents varies monotonically with change in magnetic field, and as 0 B  , they are obtained as 25/8 and 49/8 at respectively.However, singularities of MD at Fermi points, appears weaker and gradually degenerating into a smooth curve.

NN
 being the number of occupancies that a particle at the right (left) Fermi level jumps to, in the 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 particles 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, Z is the dressed charge 22  matrix describing anomalous behaviour of critical exponents.The 22  matrix elements are given by cs vv is the velocity for holon(spinon) excitations and ) conformal dimension for c  holon and s  spinon excitations.

Fk
is given by Eqn.(3.15) with critical exponent

Fig 1 .
Fig 1. Critical exponent of the momentum distribution at (a)

7 Fk
are noticed in the plots for () Gk   in Fig.4.This may be due to increase in number of particles which transfer from one Fermi level of the holon to the other.

3 Fkk and 3 Fk 7 Fk . However, singularities at 5 F k and 7 Fk 5 F k and 7
as shown in Fig.2.The calculated exponents for the MD singularities at F agrees with the results from numerical study.The calculated exponent for the MD at 5 F k is significantly 25/8 and indicates there is weak singularity near this point.This result disagrees with the numerical prediction of 1.We obtained an exponent of 49/8 at appears weaker and gradually degenerating into a smooth curve.This property at J  are integers or half-odd integer depending on the particles of the number of down and up spins, respectively.The state corresponding to the Bethe-Ansatz solution Eqns.(2.2) and (2.3) has energy and momentum given by j I and Therefore, the magnetic field dependence of the conformal dimensions Eqns.(2.6) and (2.7) are given by 1 u in the strong coupling limit, and obtained the elements to be [see Eqns.(17) to (79) of Ref. 12] 0 ( ) 1 cc Zk (2.15) 0 ( ) 0 cs Z   (2.16)