Light Cone Quantization and Savvidy Instability in Dense Quark Matter. Abstract

Transcription

1 Nisho-06/2 Light Cone Quantization and Savvidy Instability in Dense Quark Matter Aiichi Iwazaki Department of Physics, Nishogakusha University, Ohi Kashiwa Chiba , Japan. (Oct. 5, 2006) arxiv:hep-ph/ v1 10 Oct 2006 Abstract It is a longstanding problem how instability of Savvidy vacuum in gauge theory is cured. Using light cone quantization we analyze the problem not in real confining vacuum but in dense quark matter where gluons are interacting weakly with each other. Because a hermite Hamiltonian can be defined in the quantization, the problem can be solved approximately by finding ground states of gluons. We find that all of gluons in a ferromagnetic ground state carry a vanishing longitudinal momentum. This supports our previous result which states that gluons form a quantum Hall state in dense quark matter Lg, t, Aw, p, f Light Cone Quantization, Quark Matter, Quantum Hall State Typeset using REVTEX 1

2 About 30 years ago, Savvidy [1] has shown that a color magnetic field is generated spontaneously in the Yang-Mills gauge theory. Namely, when one calculates one loop effective potential of the color magnetic field, it is found that the nontrivial color magnetic field is generated spontaneously. But, soon after it has been shown [2] that some of gluons have imaginary energies under the color magnetic field and that consequently the effective potential has an imaginary part. The existence of the imaginary part implies instability of the ground state with the color magnetic field. Of course, the naive vacuum without the magnetic field is also energetically unstable. We call it as Savvidy instability. This magnetic instability of the vacuum in the Yang-Mills gauge theory was expected by many authors to lead to a confining vacuum. Namely, unstable gluons with imaginary energies are produced and may form a new stable ground state, which was expected to be a confining vacuum. The subsequent analysis [3] of the gluons has revealed complication of the color magnetic flux due to the production of the unstable gluons. Although a lattice of the flux tube has been discussed [3] to be formed, any other clear pictures of such a complicate states of the unstable gluons have not been shown. Eventually, a confining vacuum could not be obtained. To solve the problem in the vacuum we need to perform non-perturbative analysis [4]. We have recently investigated the Savvidy instability in dense quark matter and shown [5] that the instability is cured by the formation of a stable quantum Hall state [6,7] of the gluons. Since perturbative arguments such as loop expansions are applicable in sufficiently dense quark matter, the results coming from the loop expansions or perturbative analyses are reliable. Thus, we may trust the results of the spontaneous generation of the color magnetic field and the formation of quantum Hall states. The formation of the quantum Hall states is resulted from gluon s repulsive self interactions even when the interaction is small, as far as conditions of filling factor are satisfied. Consequently, Savvidy instability is cured in the dense quark matter, which is composed of quarks, the color magnetic field and the colored quantum Hall state of gluons. The phase of the matter is called as color ferromagnetic phase. Although quarks occupy Landau levels, they do not form quantum Hall states in general. ( We have shown [5] that color superconductivity [8] is realized in much denser quark matter than the matter in which the color ferromagnetic phase is realized. Thus, it is more important phenomenological than the color superconducting phase. We have discussed an astrophysical implication of the phase [9] and have pointed out the similarity [10] between the gluons in the dense quark matter and color glass condensate in nucleons. ) Quantum Hall states arise only in two dimensional space. For example, quantum Hall states of electrons are realized in quantum well of semiconductors, which is a two dimensional well. Then, it is natural to ask how two dimensional quantum Hall states of gluons are formed in the three dimensional dense quark matter. In this paper we analyze the problem as well as Savvidy instability, using the light cone quantization [11,12]. Since Hamiltonian can be well defined in the quantization, in order to answer these questions we have only to obtain ground states of the Hamiltonian with the use of an approximation valid at small couplings. As a result we find that gluons occupying the lowest Landau level are produced to form a state, in which all of the gluons have a vanishingly small longitudinal momentum. Furthermore, we find that excitations with non vanishing longitudinal momentum have finite gaps. Thus, they are two dimensional since the ground state is uniform in the longitudinal direction and there exist the finite gap in the longitudinal direction. Hereafter we analyze SU(2) gauge 2

3 theory. First of all, we will give a brief review of how the two dimensional quantum Hall states of gluons arise in the previous our analysis [5,7] with equal-time quantization. The energies E of the unstable gluons, A 1 l +ia2 l e iet, under a color magnetic field, B = ǫ 3ij i A 3 j, uniform in space is given by E 2 = k 2 gb, where g is gauge coupling constant and k is a momentum parallel to the magnetic field B = (0, 0, B). The gluons occupy the lowest Landau level. The energies of other stable gluons are given by E 2 = k 2 + gbn with integer n 0. Thus, the unstable gluons with k 2 < gb have imaginary energies and their amplitudes grow up rapidly in time. Among them, the most unstable gluons are such ones with vanishing momentum k = 0. Hence, their wave functions are uniform in the direction parallel to the magnetic field. These most unstable gluons are expected to form a new stable ground state similar to the case in Higgs model with a negative mass term m 2 φ 2. In the model, unstable Higgs modes with energies, E 2 = k 2 m 2 < 0 are present in the naive vacuum φ = 0, among which, the most unstable modes with k = 0 form a new ground state, φ 0 with the condensation of the field φ. Similarly, we expect that a stable ground state of gluons is made of only the gluons with k = 0. They are two dimensional objects since they are uniform in the direction of B. It is well known that bosons as well as fermions in two dimensional space can form a quantum Hall state if repulsive short range interaction is present. Hence, the two dimensional gluons make a quantum Hall state due to the repulsive self-interaction [6,13]. In this way the two dimensionality of gluonic ground state arises from the expectation that only the most unstable gluons with k = 0 make a stable ground state; any gluons with k 0 do not contribute to the formation of the ground state. This is our result in the equal-time quantization. In such a circumstance, we wish to examine in the light cone quantization whether ground state of gluons are two dimensional or not. In the formulation, the states with imaginary light cone energy never appear. Instead, the naive vacuum which is annihilated by all of the annihilation operators is not the lowest energy state. The real ground state with the lowest energy is formed as a composite state of gluons in the lowest Landau level. In this paper we neglect zero modes of gauge fields in the light cone quantization. We do not investigate the real confining vacuum which is a complicated composite of strongly interacting gluons; the problem of obtaining such a real vacuum is beyond our scope [4]. Instead, we investigate a ground state of gluons in dense quark matter in which gluons are weakly interacting with each other. We assume the presence of a color magnetic field, which is generated spontaneously. Here, we do not address a question of spontaneous generation of color magnetic field in the light cone quantization. Now, we show that a ground state of gluons is two dimensional, that is, it depends trivially on longitudinal momentum p +. We use the notations of the light cone time coordinate, x + = (x 0 +x 3 )/ 2 and longitudinal coordinate, x = (x 0 x 3 )/ 2. Transverse coordinates are denoted by x i, or x. We assume a finite length, L x L, in the longitudinal space and impose that gauge fields are periodic in x. Then, corresponding momentum becomes discrete denoted by p + n = nπ/l with integer n. Light cone components of gauge fields, A +, A, A i, are defined similarly. Then, the Hamiltonian, H, with the light cone gauge, A + = 0 is given by H = 1 4 F a ij F a ij + g2 2 ρa 1 ( 2 ) ρa, (1) 3

4 with transverse components of field strength, Fij a, where color indices a run from 1 to 3 and space indices, i, j run from 1 to 2. ρ a is defined by ρ a = (D i A i ) a = ( i δ ab + gǫ a3b A B i )Ab i (2) where A B i denotes the gauge potential of color magnetic field, which is assumed to direct into σ 3 in color SU(2); B = 1 A B 2 2 A B 1. We have neglected a dynamical gauge potential, A a=3 i aside from the classical one, A B i since they do not couple directly with A B i. We have only taken dynamical gauge fields, A a=1,2 i perpendicular in color space to the color magnetic field. They form Landau levels under B. We treat only the quantum effects of the gluons, but treat quarks classically for simplicity. Their color charge neutralizes the color charge of the gluons. We make a comment on zero mode in the light cone quantization. Our treatment of zero mode [14,15] is similar to the one used by Thorn [16]: We quantize gauge fields in longitudinal direction with finite length and neglect zero modes of the fields. Consequently, Hamiltonian becomes a simple form involving at most quartic terms of creation or annihilation operators in addition to quadratic ones. As being shown [16] in a two dimensional model of scalar field, the true ground state can be gripped even if we neglect the zero mode of the field, at least in the limit of L. We assume that it also holds in the gauge theory. Neglecting the zero mode may be justified in another way. That is, our concern is not the real vacuum, but a ground state of gluons in dense quark matter, in which there is no condensation of gluon fields. To treat correctly the zero mode may be essential for investigating the real vacuum of strongly interacting gluons. But, it would not be important to take seriously the zero mode for investigating a ground state of weakly interacting gluons in the dense quark matter. Typical energy scale of quarks and gluons in the quark matter is given by the chemical potential of the quarks and is much large. Thus, it is similar to the case of QCD at high energy scattering [17] where the zero mode does not play an important role. Therefore, it is reasonable to neglect the zero modes of the gluons in the quark matter. We have assumed the spontaneous generation of the color magnetic field. The condition of neglecting the zero mode requires the coherent length of B in x being shorter than L, but going to infinity as L. Thus, our discussion below is limited for the case that momentum scale, p + of gluons is larger than the inverse of the coherent length of B. In the light cone gauge only dynamical variables are transverse components, A a i of gauge fields, which can be expressed in terms of creation and annihilation operators, A b i = π L p + >0 1 2πp + (ab i,p (x+, x)e ip+ x + a b i,p(x +, x)e ip+ x ), (3) with p + = πn/l, where operators, a l i,p, satisfy the commutation relations, [a b i,p(x +, x), a c j,k (x+, y)] = δ ij δ bc δ pk δ( x y), with other commutation relations being trivial. As we have mentioned before, we have neglected the zero modes, p + = 0, of the gauge fields. Then, the gauge fields satisfy the equal time, x +, commutation relation, [ A a i (x +, x, x), A b j(x +, y, y)] = iδ ij δ ab δ( x y) ( δ(x y ) 1 ), (4) 2L 4

5 where the last factor, 1/2L, in the right hand side of the equation comes from neglecting the zero modes of the gauge fields. We should mention that the second term in H represent a Coulomb interaction. It is derived by solving a constraint equation, 2 A,a = ρ a, that is, Gauss low resulted from the light cone gauge condition, A + = 0. In order to assure that the gauge field, A is periodic in x, the zero mode of ρ ( ρ n=0,1,2, ρ n e iπnx /L ) must vanish; ρ n=0 = 0. Then, the operation of 1/( 2 ) is well defined. The condition of ρ n=0 = 0 implies the total color charge, L L dx ρ tot, vanishes. Here we should include background classical color charge of quarks in ρ tot. ρ in eq(5) should be replaced by ρ tot = ρ ρ quark with ρ tot,n=0 = 0. This requirement of ρ tot,n=0 = 0, is consistent with the condition, A n=0 = 0. We now rewrite the Hamiltonian in terms of charged vector fields, Φ i = (A 1 i +ia2 i )/ 2, which may be decomposed into spin parallel ( anti-parallel ) component, Φ u = (Φ 1 +Φ 2 )/ 2 ( Φ ap = (Φ 1 Φ 2 )/ 2 ). These fields transform as Abelian charged fields under the U(1) gauge transformation, A i U A i U with U = exp(iθσ 3 ). Then, using the fields, Φ p ( Φ ap ), we obtain the following Hamiltonian, with D = + ig A B, where ρ is given by H = 1 2 B2 + Φ p ( D 2 2gB)Φ p + Φ ap ( D 2 + 2gB)Φ ap (5) + g2 2 ( Φ p 2 Φ ap 2 ) 2 + g2 2 ρ 1 ( 2 ) ρ, ρ = i(φ p Φ p Φ p Φ p + Φ ap Φ ap Φ ap Φ ap) ρ quark. (6) The first term represents the classical energy of the color magnetic field, and second ( third ) term does the kinetic energy of the charged gluons with spin parallel ( anti-parallel ) under the color magnetic field, B. The forth term represents the energy of the repulsive self-interactions. The last term represents the Coulomb energy coming from the second term with ρ a=3 component in eq(5). Since eigenstates of the operator D 2 are classified by Landau levels, the second and the third terms can be rewritten as, ( Φ p, n (2n 1)gBΦ p,n + Φ ap, n (2n + 3)gBΦ ) ap, n, (7) n=0,1,2,,, where the fields, Φ p, n ( Φ ap, n ) denote operators in the Landau level specified by integer n. ( We have implicitly assumed integration over the transverse directions. ) Now, we take only the field, Φ p, n=0 in the lowest Landau level, n = 0, that is, the component having negative kinetic energy. It is most important, among others, for realizing the ground state of the Hamiltonian in the limit of strong magnetic field, B. It corresponds to the unstable gluon in our previous discussions [5]. Therefore, we consider the following reduced Hamiltonian for exploring the ground state of the system, H r = gb Φ 2 + g2 2 Φ 4 + g2 2 ρ 1 r ( ) ρ r, (8) 2 5

6 with ρ r i(φ Φ Φ Φ) ρ quark, where we have put Φ Φ p, n=0 for simplicity. The field, Φ, can be expressed by using creation and annihilation operators, π 1 Φ = (a p,m φ m ( x)e ipx + b L 2πp p,m φ m ( x)eipx ), (9) p>0,m=0,1,2,, where simplified notation such as x = x and p = p + is exploited and will be used below. φ m ( x) = g m z m exp( z 2 /4l 2 ) represents normalized eigenfunction of D 2 with angular momentum, m, in the lowest Landau level, d 2 xφ m φ n = δ m,n ; z = x 1 + ix 2 and g m 1 πm!(2l 2 ) m+1 with l 2 1/gB. a p,m and b p,m satisfy the commutation relations; [a p,m, a k,n ] = δ p,kδ m,n, [b p,m, b k,n ] = δ p,kδ m,n, others = 0. When we express the first term in eq(8) in terms of the operators, a p,m and b p,m, L L dxd 2 x : gb Φ 2 := gb p>0,m 1 p (a p,m a p,m + b p,m b p,m), (10) we find that there exist states with lower energies than the trivial vacuum, vac >; a p,m vac >= b p,m vac >= 0. Namely, gluons occupying the lowest Landau level are produced spontaneously to form a state with energy lower than that of the vacuum. The production of the gluons are limited by the second term in eq(8) representing repulsion among the gluons. This is similar to the case of Higgs model. Contrary to the model, the gluons do not condense. Thus, we postulate Φ = 0. ( This is consistent with neglecting zero mode. ) In order to find the ground state of the gluons, we will express approximately the energy, H r, of the state in terms of a p,m a p,m and b p,m b p,m, with the condition, a p,m = b p,m = 0. Then, by minimizing the energy we will find the momentum, p, and angular momentum, m, distribution of gluons. Such gluons may carry non vanishing color charge density, in general. The fact that quarks and gluons are weakly interacting in the dense quark matter certificates that our approximation gives rise to reliable results; any quantum corrections to the results are small. Before finding the ground state of the Hamiltonian, H r, we should note that there are two conserved quantities such as total color charge and momentum, Q = L L P total = p>0,m dxd 2 xρ r = p>0,m (a p,m a p,m b p,m b p,m) L L dxd 2 xρ quark = 0 p(a p,ma p,m + b p,mb p,m ), (11) where we have required that the total color charge of gluons and quarks must vanish. The ground state must be found under the condition of the conserved quantities being given. Since our concern is the ground state of gluons in dense quark matter, not real vacuum, the color charge of the gluons may be non vanishing and is neutralized by the color charge of quarks. In order to evaluate the expectation value of H r with the ground state, g, we assume the following approximation, a α a β a γa δ a α a γ a β a δ + a α a δ a β a γ, forα β, similar for b α (12) a αb β a γb δ a αa γ b β b δ, a a b a = a b b b = a aab = b bba = 0, (13) a α a β δ α,β, b α b β δ α,β (14) 6

7 where indices, α, β,,, denote a set of p and m. Namely, we assume no mixing among states with different p and m in eq(12). Furthermore, we assume no pair creations and annihilations in eq(13) so that the numbers of particle and antiparticle are conserved, respectively. We also postulate momentum and angular momentum conservation in eq(14). Using the approximation, we evaluate an expectation value, : H r :, of normal ordered Hamiltonian, : H r : and wish to find the ground state minimizing the expectation value, : H r : = gb ( ) a(p, m) + b(p, m) + p>0,m + g2 2L + g2 2L + g2 2L p,q>0,m,n p q>0 ( a(p, m)a(q, n) + b(p, m)b(q, n) + 2a(p, m)b(q, n) ) Nm,n ( ) p + q 2 ( ) a(p, m)a(q, n) + b(p, m)b(q, n) Nm,n p q m,n ( ) p q 2 2a(p, m)b(q, n)n m,n, (15) p + q p,q>0 m,n with N m,n = (m + n)! (πm! n! 2 m+n+2 l 2 ) 1, where a(p, m) and b(p, m) are defined by a p,m a q,n = δ p,q δ m,n p a(p, m), b p,m b q,n = δ p,q δ m,n p b(p, m), (16) where a(p, m) and b(p, m) represent distributions of p and m in the ground state. We have used the color neutrality condition, ρ r,n=0 = 0 in the derivation of eq(15); the condition is imposed as a constraint, p q in the third term. We find in the approximation that the expectation value of the color charge density becomes uniform in longitudinal direction, x = x, ρ r = 1 L p>0,m p ( a(p, m) b(p, m) ) φ m ( x) 2 ρ quark, (17) which may depend on transverse coordinate, x. In eq(15) the first term represents the kinetic energy in the Landau level and the second term does the energy of the repulsion between gluons. They are denoted by E 1. The third and forth terms represent the Coulomb energy between gluons denoted by E 2. Obviously, these terms are non negative except for the first one. Therefore, we can find a ground state with the lowest energy, : H : E 1 +E 2, by minimizing the first two terms, E 1 and the last two terms, E 2 of the Coulomb energy, respectively. Simultaneously, we need to take into account the condition that the state carries the given total momentum and vanishing total color charge. We may assume that the color charge of quarks, ρ quark is spatially uniform. It is easy to minimize the first two terms, i.e. E 1. The result is given by a set of values, c(m) p>0(a(p, m) + b(p, m)) since E 1 can be rewritten such as E 1 = gb m c(m)+g 2 /(2L) m,n c(m)h m,n c(n). Thus, minimizing the energy, E 1, does not determine the dependence of the state on the longitudinal momentum, p = p +. It simply gives c(m) gbl/g 2. The dependence is determined only by minimizing the Coulomb energy, E 2. The energy, E 2 0, can be minimized by assuming that the ground state depends only on a single momentum, p = p 0, that is, a(p, m) δ p,p0 and b(q, n) δ q,p0. This distribution of the momentum leads to the minimum, E 2 = 0. ( Any other distributions with the dependence on 7

8 various momenta have higher energies ( > 0 ). ) It turns out from the solution that the color m( charge density is given such that ρ r = p 0 a(p0, m) b(p 0, m) ) φ m ( x) 2 /L ρ quark = 0. For the color charge density to vanish, a(p 0, m) b(p 0, m) should be independent of m, since m φ m ( x) 2 = 1/(2l 2 π). Then, the color charge density of gluons can cancel that of quarks, ρ quark. On the other hand, the total momentum can take any positive value since total momentum is given by P tot = m p 2 0(a(p 0, m) + b(p 0, m)) = p 2 0 m c(m). Therefore, the ground state minimizing the energy in eq(15) is characterized by the trivial dependence of the longitudinal momentum; a p,ma p,m δ p,p0 and b p,mb p,m δ p,p0. The Coulomb energy, E 2, vanishes in the state. All of gluons in the lowest Landau level occupy the states with a single momentum, e.g. p = p 0. It implies that the ground state is formed by two dimensional gluons occupying the lowest Landau level. It is also important to note that displacement of a gluon with the momentum, p 0 in the ground state to a state with a momentum k p 0 gives rise to an energy gap, E k ; k : H r : k = : H r : + E k ( E k > 0 for any k ) where k a k,m a p 0,m g. It is easy to see that the finite gap results from the Coulomb interaction. We note that k as k p 0. This simple argument suggests the existence of a finite gap energy needed to excite modes in the longitudinal direction. Then, gluons can form two dimensional states as far as their energies are less than the gap energy. More detail analysis is necessary to convince the conclusion. We wish to mention that for P total to be finite in the limit of L, p 0 goes to 0 in the ( limit because c(m) L. On the other hand, the color charge density of gluons, p 0 a(p0, m) b(p 0, m) ), can take any finite value even in the limit. Hence, all of gluons occupy the states with a vanishingly small longitudinal momentum. In the terminology used in high energy scattering of hadrons or nuclei, parton distribution in the dense quark matter becomes such as δ(x) since x = p 0 /P tot 0. ( The distribution contains only contributions of charged vector field, Φ ± or A a=1,2 i. If we take into account momentum distribution of the gauge field, A a=3 i neglected in our argument or quantum corrections of Φ to the distribution, they would give a parton distribution with wide range in x. ) We have not yet shown that the gluons forms quantum Hall states, but have only shown they forming two dimensional states. It is a necessary condition for the realization of quantum Hall states. Whether or not the quantum Hall states of gluons are made, depends on the color charge density of the gluons. If a specific condition of filling factor, ν = 2πρ r L/gB is satisfied, the quantum Hall state can be realized, that is, m( ν = 2πp 0 a(p0, m) b(p 0, m) ) φ m ( x) 2 /gb = 1/2, 1/4,,,. Otherwise, gluons simply form a two dimensional compressible state, i.e. gapless state. As we have shown, the two dimensionality of the ground state of gluons arises from the Coulomb interaction, ρ 1 2 ρ in the longitudinal direction, x. In our approximation, the Coulomb energy, E 2 is a positive semi definite so that the ground state is given such as the state with E 2 = 0. When we apply the similar approximation to quarks coupled with the magnetic field, we obtain a negative semi definite Coulomb energy. Thus, we do not obtain the two dimensionality in the quarks. In the ground state of the quarks, the distribution of the longitudinal momentum is never of delta function type. The result in the quarks or fermions is physically natural. Electrons in metals never form two dimensional gases under external magnetic field except for ones confined in two dimensional space, e.g. quantum wells. The difference between the case of gluons and that of quarks comes from their difference 8

9 in statistics. In order to see this point more explicitly, we take a following simple model of a Coulomb interaction, V c = g2 2 : ρ 1 ( 2 ) ρ : ρ(x) = ψ (x)ψ(x) ψ ψ (18) with L x L, where the field, ψ represents boson or fermion with an appropriate boundary condition at x = ±L. Suppose that we have two states with momentum, p > 0 and q( p) > 0, whose wave functions are denoted as f(p, x) and f(q, x) e iqx. We define the field operator such that ψ(x) = a p f(p, x) + a q f(q, x), (19) with annihilation operators, a p and a q satisfying commutation relations, [a p, a q] ± = δ p,q,,. As in the previous case, we extract zero modes of ρ for 1/( 2 ) to be well defined. Thus, ψ ψ = a p a qf (p, x)f(q, x) + h.c. When we evaluate the expectation value of V c with the use of a state, a pa q 0, V c ± = ±2 f(p, x) 2 f(q, x) 2 1 (p q) 2, (20) we find that the Coulomb energy is positive semi definite in the bosonic case, V c + 0, and negative semi definite in the fermionic case, V c 0. If we allow additional internal states with an identical momentum, i.e. f i (p, x) with i = 1, 2 and evaluate V c with the use of a state, a p,i=1a p,i=2 0, we find V c = 0. Therefore, it turns out that in the case of bosons, the state of two particles with an identical momentum is more stable than the state of two particles with different momentum. On the other hand, in the case of fermions, the situation is reverse; the state of two particles with different momentum is more stable than the state of two particles with an identical momentum. To summarize, we have shown using light cone quantization that Savvidy instability is cured in sufficiently dense quark matter where quarks and gluons couple weakly with each others. In the quark matter the color magnetic field is generated spontaneously and gluons occupying the lowest Landau level are also produced spontaneously to form a ground state of gluons. All of gluons in the ground state carry a vanishing longitudinal momentum. Thus, they form a non trivial two dimensional state with non vanishing color charge. The state arises due to the effect of Coulomb interaction among bosonic gluons in longitudinal direction. We need further analysis to see whether or not these gluons form quantum Hall states. We would like to express thanks to Prof. O. Morimatsu, Dr. T. Nishikawa and Dr. M. Ohtani for useful discussion. This work was supported by Grants-in-Aid of the Japanese Ministry of Education, Science, Sports, Culture and Technology (No ). 9

10 REFERENCES [1] G.K. Savvidy, Phys. Lett. B 71 (1977) 133. [2] N.K. Nielsen and P. Olesen, Nucl. Phys. B 144 (1978) 376; Phys. Lett. B 79 (1978) 304. [3] J. Ambjørn, N.K. Nielsen and P. Olesen, Nucl. Phys. B152,75 (1979). H.B. Nielsen and M. Ninomiya, Nucl. Phys. B156, 1 (1979). H. B. Nielsen and P. Olesen, Nucl. Phys. B160, 330 (1979). [4] K. Kondo, Phys. Lett. B600, 287 (2004). [5] A. Iwazaki and O. Morimatsu, Phys. Lett. B 571 (2003) 61. A. Iwazaki, O. Morimatsu, T. Nishikawa and M. Ohtani. Phys. Lett. B 579 (2004) 347. A. Iwazaki, O. Morimatsu, T. Nishikawa and M. Ohtani, Phys. Rev. D 71 (2005) [6] The Quantum Hall Effect, 2nd Ed., edited by R.E. Prange and S.M. Girvan ( Springer- Verlag, New York, 1990 ). S. Das Sarma and A. Pinczuk (Eds.), Perspectives in Quantum Hall Effects, (Wiley, New York, 1997). [7] Z.F. Ezawa, M. Hotta and A. Iwazaki, Phys. Rev. B46, 7765 (1992); Z.F. Ezawa and A. Iwazaki, J. Phys. Soc. Jpn. 61, 4133 (1990). [8] K. Rajagopal and F. Wilczek, hep-ph/ [9] A. Iwazaki, Phys. Rev. D72 (2005) [10] A. Iwazaki, hep-ph/ [11] P.A.M. Dirac, Rev. Mod. Phys. 21 (1949). [12] for a review, see A. Harindranath, hep-ph/ , T. Heintl, hep-th/ [13] T. Nakajima and M. Ueda, Phys. Rev. Lett (2003). [14] T. Maskawa and K. Yamawaki, Prog. Theor. Phys. 56 (1976) 270. [15] S. Tsujimaru and K. Yamawaki, Phys. Rev. D57 (1998) [16] J.S. Rozowsky and C.B. Thorn, Phys. Rev. Lett. 85 (2000) [17] E. Iancu and R. Venugopalan, hep-ph/