- Email: [email protected]
Unitary polynomials in normal matrix models and wave functions for the fractional quantum Hall effects
PhysicsLettersA 167 (1992) 452—458 North-Holland
PHYSICS
E L TTERS A
Unitary polynomials in normal matrix models and wave functions for the fractional quantum Hall effects Ling-Lie Chau Department of Physics, University ofCalifornia, Davis, CA 95616, USA and CNLS. Los Alamos National Laboratory, LosAlamos, NM 87545, USA
and Yue Yu Department of Physics, University ofCalifornia, Davis, CA 95616, USA and Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China Received 23 October 1991; revised manuscript received 4 June 1992; accepted for publication 4 June 1992 Communicated by D.D. Hoim
We formulate matrix models with random interactions from normal matrices and construct the corresponding unitary polynomials. The unitary polynomials can be viewed as a set of one-body wave functions of a two-dimensional repulsive Coulomb gas of particles. We then construct the N-body ground state wave functions from these unitary polynomials according to symmetry and statistics requirements. Interestingly, in the case of the zero matrix-model random interaction we precisely obtain the Laughlin wave functions for the fractional quantum Hall effect. In the case of the nonzero matrix-model random interactions, we demonstrate that the N-body wave functions obtained from our normal matrix model describe electrons on a plane under the influence ofnonuniform external electromagnetic fields. Thus, our solutions from the normal matrix model show that fractional quantum Hall effects do exist in complex situations of external electromagnetic fields. It will be interesting to observe such fractional quantum Hall effects experimentally.
Recently, there has been much interest in the relationships between various two-dimensional theories. The rich structures of the ground states in strongly interacting electron systems [1] may be closely related to two-dimensional conformal field theory as well as to the (2 + 1)-dimensional Chern— Simons gauge theory, Abelian and non-Abelian [2]. Another example recently pointed out by Callaway [3] is that the Laughlin wave functions [4] for the fractional quantum Hall effect (FQH) [5] can also be constructed from the point of view of a random matrix model [6]. Matrix models are known to provide a powerful tool for calculating nonperturbative effects in two-dimensional gravity [7,8]. This new connection between the matrix models and the FQH theory is very fascinating, In this paper, we formulate matrix models for the normal matrices and construct the unitary polyno452
mials in the normal matrix models with arbitrary orders of matrix-model random interaction. We cornpare the unitary polynomials to the orthogonal polynomials in the Hermitian matrix models [9] and obtain many identity relations for the unitary polynomials. It is well known that wave functions describing the one-dimensional repulsive Coulomb gas of particles can be constructed from the orthogonal polynomials of the Hermitian matrix models [9]. We show that wavefunctions describing the two-dimensional case can be constructed from the unitary polynomials given here for the normal matrix model. The use of the normal matrix is essential. Since the matrix and its Hermitian conjugate commute, i.e., [M, M~J=0 for a normal matrix M, they can be simultaneously diagonalized. The complex eigenvalues of the nor-
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 167, number 5,6
PHYSICS LETFERS A
mal matrices can be interpreted as the positions of particles in two dimensions. Taking the unitary polynomials of the normal matrix model as set of one-body ground state wave functions in two dimensions, we can construct the Nbody wave functions ofthe ground state uniquely according to the rules in the matrix model and the U (1) symmetry. In the case of zero matrix-model random interactions, we obtain exactly the Laughlin wave functions for the FQH effects of electrons moving in a two-dimensional plane under the influence of a uniform external magnetic field perpendicular to the plane. This is an interesting coincidence. We next investigate the situation of non-zero matrix-model random interactions. Indeed our exact solutions from the normal matrix model provide new results for the FQH effects. We show that the N-body ground state wave functions obtained describe electron systems on a plane influenced by complicated electromagnetic fields. We demonstrate explicitly that the leading order matrix-model random interactions (g1 0) describe a nonuniform magnetic field perpendicular to the plane and a nonuniform static electric field in the radial direction. We then show the existence of the fractional quantum Hall effects and calculate the filling factor. It will be interesting to observe such effects experimentally. Let M be an Nx N normal matrix, [Mt, M] = 0, with the distinct eigenvalues ;=x1+ iy~where Xj and y~(j= 1, N) are real. Quantum mechanically, our normal matrix models are described by the partition function ...,
Z(g)=
J
~11dM~ exp[ —Tr V(M)],
(1)
where V(M)=IMtM+ n~~ ~ 2N
3 August 1992
to a constant factor, the partition function Z(g) is [61 r N Z(g) j fl dji(~) 4 (z1, ZN) 2, (3) ...,
where du (z~)= d2~d~exp [ V(~)= ~ZJZJ =
—
V(z3)],
g~_1
-
-
n~2 N”’ Z7 Z7
+
I 12 + ~
I I 2n
~
(4)
n~2 N
and 4(z1, 4(z,,
...,
...,
ZN) 15
ZN) =
the Van de Monde determinant
[II
(Z1—~j)
(5)
.
>‘
There is a global U (1) symmetry in eq. (3) under (
,~
~,
~
1
which is equivalent to the symmetry of eq. (1) under M—~UM. From the partition function (3), as done in ref. [9] for the Hermitian matrix models, we can construct a set of polynomials, which we call unitary polynomials. When the dimension of the matrix reduces to one, the partition function represents the probability with the weight exp [ —Tr V(Z)]. This fact means that the unitary polynomials must be of this weight. By defining
r
j fl
~
d~(z,)J4(Z1,
...,
(7)
Z,,) 2,
which is equal to the partition function Z of eq. (3) when n = N, we then construct a set of polynomials of degree n, 2
(2)
and terms with coupling constants g~describe the matrix-model random interactions. The theory is invariant under M-+ UM, where UU~=1, while [M, U] = 0 is required in order that UM is also a normal matrix. Since Mis a normal matrix, there exists a unitary matrix S which can simultaneously diagonalize M and Mt to ZD= (d~~z1) and ZD= (d~j2~) respectively: M= StZDS and Mt = S45t. Then, up
Pfl(Z)=Z;1(g)Jfld4u(ZJ)
X
(z— Z~)
.
I4(Zi,...,z~)I (8)
The leading term in the power of z is Z” with the coefficient unity. Similar to those identities found in the Hermitian matrix models [9] we can obtain the following identities,
453
Volume 167, number 5,6
PHYSICS LETTERS A
n+I
‘
I fl
d~i(~) 4(Z 1,
•‘ j=1
Z~+l)4*(Zl,
...,
...,
2+.... (16) Because the ~ are complex (bnm ~ 0, m < n 1) the P~(Z)=z”+c~,~_2z~ recursion relations among the unitary polynomials
z~)Z~,
—
n +1
I’
=(
JJ d~z(~)4(Z1,...,Z~~,)
1)n+1_k
~ X 4* (Z1,
...,
J
)~
Z~, ...~ ~
(9)
are no longer as simple as those between the orthogonal polynomials in the Hermitian matrix models. For the latter, only three polynomials with orders of n + 1, n and n 1 appear in the recursion relation, which is important in order to derive the string equations by means of the orthogonal poly—
Z~ ~
~
~
~ n+I
j {J dy(~)
=
3 August 1992
4(Z,,
...,
Z~ i)4*(Z,,
...,
Z~)Z~÷,
nomials [7,8]. So the string equations in the normal matrix model has to be derived in a different way Since h~=Z~÷,/(n+l)Z~, Z~can be representedby h~, (17)
Z~=n!h~_,...h,h0,
x ~ (_l)n~_kA*(Zl,...,~k,...,Zfl)z~, (10) k= i 2k
represents the absence of Zk in
where the symbol 4*in eqs. (9) and (10). Itiseasyto seethatthesum in eq. (10) is the complex conjugate of the determinant ~1 1
z, z2
1
...
Z
Zr’
Z~
zr+{
Z~÷,
I
Z?’
which is zero for s= 1,
J
~1)
...,
n
Zn +
1
1. For s = n, we have
—
d/2(Z) P~(Z)P~(Z) hndm,n,
h
=
h~are
(12)
(13)
real, (14)
Zfl+1
(n + 1) Zn Consider the following expansion in the unitary polynomials, —
n+l
ZP~(Z)
=
~
~~P1(Z)
exp[—~V(z)]
,
(18)
Jcb~~ØidZdZ=d/m.
(19)
Z~+j
Hence, the polynomials P~(Z) are complex orthogonal, we call them unitary polynomials,
where
,=
which satisfy the unitary condition
I
Z~ dp(~)P~(z)Z”= n+l~
J
but the relation between Z~and ~ is not as simnomials P in the orthogonal polynomials. The polyple as that 1(z) may be normalized as
b,,,,, Pm (z).
(15)
This concludes our matrix-model formulation for normal matrices. We emphasize that the unitary polynomials constructed are for arbitrary matrix-model random interactions as given in eq. (2). Next we shall calculate N-body wave functions of the normal matrix model using the unitary polynomials derived above in the case of zero matrix-model random interactions; i.e. g~=0and only the Gaussian first term in eq. (2) exists. Very interestingly they precisely give the Laughlin wave function for FQH effects. We investigate the case of non-zero matrixmodel random interactions. We demonstrate that the N-body wave functions obtained from our normal matrix model describe electrons on a plane under the influence of nonuniform external electromagnetic fields, for which we also show the FQH effects exist and calculate the filling number. Thus the normal matrix model provides new insights for FQH effects
m=O 1*1
Making use of eq. (13), we have b~~÷1 = 1 and = 0, i.e. the unitary polynomials have the form 454
For a general complex matrix model, the string equation has been discussed in ref. [10], which contains the string equation for the normal matrix model discussed here.
Volume 167, number 5,6
PHYSICS LETTERS A
from which we illustrate that fractional quantum Hall effects do exist in quite complex situations of external electromagnetic fields. Let us now briefly describe the physical system of the FQH. In the FQH, the N-electron system is restricted to an (x, y)-plane and the spin of the electrons is frozen by a uniform external magnetic field, B, perpendicular to the plane. The one-electron Hamiltonian in the FQH is H,=_~i__(~V+~A”1, 2me\i C J
(
1k Z ~t
1,
...,
Zn)
N
=
fl
/ (~,—~)~“~‘
exp( \
1>J
N —~
itary polynomials 0, of eq. (18) are the one-body ground state wave functions for the one-electron Hamiltonian H1, H~ Oi = I hw~0,.
~ 1Z112)
(21)
‘=‘
for the Hamiltonian where the background and the Coulomb interactions u between N electrons are small, ~ H~(~) + u.
(22)
(24)
Since the set of On is complete, the following functions may be expanded in 0,, 0~:::~,raP~(Z)...P~,(z)exp[—IVo(Z)1
(20)
where A is the electromagnetic potential of an external magnetic field B. In order to write down the Schrödinger equation explicitly, usually the symmetric gauge, A= ~B(y~—x9) is used. The important result [4,5] is that the variational ground state wave functions of N-body electrons are
3 August 1992
=~a,Ø,,
(25)
where a...b and n...m are arbitrary positive integers and the a, are complex constants. It is easy to check that O~:::~ are also the one-body ground state wave functions. From On we then construct the N-body ground state wave functions, eq. (21), from the point of view of the normal matrix model. Note that the partition function, eq. (3), for g~= 0 is exactly the probability associated with the Laughlin wave function Y’k=~o. Then,becausethereisthenormoftheVandeMonde determinant in the measure of the integration, all Nbody wave functions in the frame ofthe matrix model must carry the determinant factor. The U (1) symmetry of the theory requires that the N-body wave functions are U (1) covariant, which implies homogeneity of these functions in the z3. This requirement coincides nicely with the requirement of definite angular momentum for the N-particle system,
Therefore Hwk(zI,
...,
ZN)=
[INhwC+O(u)]~fk(Zl,
...,
ZN),
i.e. the wave function is an eigenfunction for the angular-momentum operator
(23) .whére the magnetic has been taken uni2=l,length and wc=eB/mec is to thebecycloty, l=(hc/eB)” tron frequency. The ground state energy is lifted by 0(u) ~ In order to use the normal matrix models to discuss the FQH, we identify the matrix M of our matrix model as the position operator for the N electrons and the complex eigenvalues of M as the twodimensional positions ofthe electrons. Thus it is natural to use normal matrices so that the position operator matrix Mand its Hermitian conjugate Mt can be diagonalizable simultaneously by a unitary matrix. We then observe that for g~=0, V(Z)— V 2), the normalized set of the un0(Z) ~exp(—~ 1Z1 .~
N ~
(~ ~
— ~
Moreover, since we are considering an N-fermion system, we require the N-body wave functions cornposed from the one-body wave functions, after factoring out the Van de Monde determinant that is totally anti-symmetric in (Z~, ZN), to be symmetric. The unique N-body ground state wave functions satisfying the above requirements are ...,
455
Volume 167, number 5,6
PHYSICS LETTERS A
Nmec
2k
~
where ~ is the unit radial vector in the (x, y)-plane and ~ is perpendicular to the plane. The vector po-
e’J’Po(ZI)PI(zi)...PN_l(Zl)) I
x exp(
—
I
~
V0
(~5
))~
(26)
where ~ is + 1 for even permutations of 1, N and —1 for odd ones. The sum of i,j, /in eq. (26) is nothing but the determinant ‘~“
...,
tential A and the scalar potential ~‘ in the symmetric gauge are A = IB0( 1+2 ~
...,
det[Pfl_l(~)]=det(z7’)=A(ZI,...,ZN). (27) The wave function (26) changes covariantly by exp[~iN(N—l)(2k+l)O]whenthez 3transformas given by eq. (6). Equation (26) is precisely the Laughlin wave function Wk, eq. (21), with arbitrary k. When there is a quasiparticle excitation in the system, the Laughlin treatment [4,5] for this can be used. First, a factor Z Z0 where Z0 is the position of the quasiparticle, is multiplied with 0~:::~ of eq. (25), except 02:::~.Because of the completeness of the set of 0’s, the product is also a one-body ground state wave function. Then, according to our construction, the resulting N-body wave functions are —
N j=’
x exp(
—
2”~’
U (~—Zo)[4(Zl,...,zN)]
Wk(Zl,,ZnZO)
~
(30)
Nmec The Schrödinger equation for the electron is ~°
H~(g~)~v=Eyi, where H, (g1) =
~
~-~—
(3la)
(~
2
0~+ A.,) ~
—
eq,
(31 b)
j=x,y i
with A1 and ~ given by eq. (30). After some calculations one can show that the ground state wave function of eq. (31) is just ~, =
I
p, exp [ — V( Z, g,)]
,
(32a)
where V(Z, g,) and is V(z) of eq. (4) with g~~tO and g~=0forn>l (32b)
(Z5))~
(28)
which is U (1) covariant, corresponding to the coyariance of the determinant det(M—Z01) under M-+ UM. This wave function describes the quasiparticle excitation in the system with the charge e/ (2k+ 1) [11]. After establishing that in the case of zero matrixmodel random interactions the unitary polynomials of the normal matrix model can give Laughlin’s wave function, we are interested in finding out what new physical phenomena our normal matrix model describes in the cases of non-zero matrix-model random interactions, Consider the case of an electron system confined to the (x, y)-plane under the influence of the fields B(r) = _B
456
r2) ~
4g hB0 r
V 0
0(l +
(29)
E(r)= S~ihBo~p
Wk(ZI,...,ZN)=A(ZI,...,ZN)
I,)
3 August 1992
~-
r2)z,
Thus the nonvanishing leading order matrix-model random interaction, g, ~ 0, describes influences of nonuniform electromagnetic fields of eq. (29) on charged particles in the (x, y)-plane. Next we can construct the N-body wave functions for this g, ~ 0 case similar to what we did in the case of zero matrix-model random interaction. The variational ground state for the N-body Hamiltonian N
H(g, ) = ~ H, (z.,, g, ) + u
(33)
i=’
is taken to be ~vk(Zj, ZN, g,) as given by eq. (26) except that V0 is replaced by V(g1). The square of the modulus of ~‘~(g,) can be written as a classical Boltzmann distribution 2=expE—fl~eff(Z1,...,zN)], (34) ...,
IW(Zl,...,ZN)I
where l/fl=2k+ 1 is the fictitious temperature and
Volume 167, number 5,6
PHYSICS LETTERS A
3 August 1992
2
the effective potential cPeff=2(2k+l)2 ~ ln~z
a~~_O, axY=2k+lh. 1 e
(40)
1—~
I
2+
4).
+ (2k+ 1) ~ (i.~1 This effective potential describes the following physical situation: the first term in eq. (35) represents the repulsion between particles of charge 2k+ 1 via the two-dimensional Coulomb interaction; the second term is the attraction of these charges to the or2/ igin due to a charge density Po (1 /27t) (1 + 8g,r N), i.e., = —2 (2k+ 1)2
+2(2k+ 1)
~
k
J
~ ln I Z
—
‘<3
dZ2
Po( IZI
I
(k)
Po = 2k+l 1 eB(r) 2k+l hc
— —
Pe
bitrary interactions. We cannormal interpretmatrix-model these unitaryrandom polynomials as a set
) lnIZ—Zk I
Thus, the electron density in state
~1Ik(g,)
.
(36)
is equal to (37)
where B( r) is the value of B( r) ofeq. (29). In other words, the state wk(~I) has the filling factor v = (1 + 8g 2/N) / (2k+ 1), which has a variation in 1r the radial direction. To see the physical implication corresponding to such an electron density and filling factor, we apply a uniform electric field E 0 in the (x,y)-plane and analyze the situation classically. Using complex variables for vectors in the plane and setting z= X+ ~Y, v= E, =E0 + E( r) =E~+iE~,the Newton—Lorentz equation for the electron is ~,
eE~—La) m
— —
+
~
(38)
Solving the equation, the electron drift velocity arising from the electromagnetic field is eE, 2/N) +0(E?). (39) imw~(l+8g,r In the linear approximation eq. (39) is consistent with the relaxation-time approximation in transport theory. Since the electric currentj= —peevd, using eq. (37) and considering the linear approximation, we have Vd =
—
This impliesmagnetic that the FQH exist when the perpendicular field effects and the applied electric field are not uniform. This is a new physical conclusion provided by our normal matrix model with g 1 #0 and g,, ~ 2 = 0. (More complicated physical situations are implied by the general gn #0 cases.) This implies that FQH effects can exist in rather complicated situations of external electromagnetic fields. It is important to verify this experimentally. In conclusion, we conclusion have constructed the normal matrix models and their unitary polynomials for ar-
of one-body wave functions of a two-dimensional repulsive Coulomb gas. We can then also construct an N-body ground state wave function. For vanishing matrix-model random interactions we obtain Laughlin’s for theonfractional quantum Hall effect wave wherefunction the electrons the two-plane are under the influence of a uniform magnetic field perpendicular to the plane. For nonvanishing matrixmodel random interactions, the normal matrix models provide a useful way to construct N-body ground state wave functions associated with more complicated electromagnetic fields. We explicitly demonstrate that the first term of the matrix-model random interactions (i.e., the g~#0 term) corresponds to the situations of a nonuniform external magnetic field perpendicularto the plane and a static electric field in the plane. We also show that the fractional quantum Hall effect exists in these complicated electromagnetic fields. We also can calculate the filling factor. It will be interesting to observe these effects experimentally. This work was supported in part by the US Department of Energy (DOE) and INCOR of the University of California. Y. Yu was also supported by the Foundation of National Natural Science of China and the Foundation for Young Researchers in the Academia Sinica and the Grant LWTZ-1298 of the Chinese Academy of Sciences. He would like to thank the Physics Department of the University of California at Davis for the warm hospitality.
457
Volume 167, number 5,6
PHYSICS LETTERS A
References [1] F.D.M. Haldane, Phys. Rev. Lett. 51(1983) 605; B. Halperin, Phys. Rev. Lett. 52 (1984) 1583; X.G. Wen, Int. J. Mod. Phys. B 2 (1990) 239; Phys. Rev, B 40 (1989) 7387; X.G.Wen and Q. Niu, Phys. Rev. B 41(1990) 9377; B. Blok and X.G. Wen, Princeton preprint IASSNS-HEP90/23 and 55, to appear in Phys. Rev. B; LASSNS-HEP- 90/ 66; N. Read, Phys. Rev. Lett. 65 (1990) 1502. [2] R. Jackiw, Phys. Rev. D 29 (1984) 3975; J. Frohlich and T. Kerler, Zurich preprint ETH-TH/90-26; Nucl. Phys. B 354 (1991) 369, and references therein; X.G. Wen, Phys. Rev. Lett. 66 (1991) 802; Princeton preprintlASSNS-HEP-91/20; J. Fröhlich and A. Zee, Nucl. Phys. B 364 (1991) 517; A. Zee, preprint NSF-ITP-9 1-41. [3]D.J.E.Callaway,Phys.Rev.B43(l991)8641. [4] RB. Laughlin, Phys. Rev. Lett. 53 (1983) 722. [5] R.E. Prange and SM. Girvin, eds., The quantum Hall effect (Springer, Berlin, 1988). [6] M. Metha, Random matrices and the statistical theory of energy levels (Academic Press, New York, 1967).
458
3 August 1992
[7] V.A. Kazakov, Phys. Lett. B 150 (1985) 282; F. David, Nucl. Phys. B 257 (1985) 45; V.A. Kazakov, I. Kostov and A. Migdal, Phys. Lett. B 157 (1985) 295; D. Weingarten, Phys. Lett. 890 (1980) 280. [8] E. Brézin and V.A. Kazakov, Phys. Lett. B 236 (1990) 114; M.R. Douglas, Phys. Lett. B 238 (1990)176; MR. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; D.J. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127;
Nuci. Phys. B 340 (1990) 333; T. Banks, M.R. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B 238 (1990) 279. [9] E. Br&in, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59 (1978) 51; C. ItzyksonandJ.B. Zuber,J. Math. Phys. 21(1980)411; D. Bessis, Commun. Math. Phys. 69 (1979) 147. [10] J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B251 (1990) 517. [11] RB. Laughlin, in: Springer series in solid state science, Vol. 53, eds. G. Bauer, F. Kuchar and H. Heinrich (Springer, Berlin, 1984).
PHYSICS
E L TTERS A
Unitary polynomials in normal matrix models and wave functions for the fractional quantum Hall effects Ling-Lie Chau Department of Physics, University ofCalifornia, Davis, CA 95616, USA and CNLS. Los Alamos National Laboratory, LosAlamos, NM 87545, USA
and Yue Yu Department of Physics, University ofCalifornia, Davis, CA 95616, USA and Institute of Theoretical Physics, Academia Sinica, P.O. Box 2735, Beijing 100080, China Received 23 October 1991; revised manuscript received 4 June 1992; accepted for publication 4 June 1992 Communicated by D.D. Hoim
We formulate matrix models with random interactions from normal matrices and construct the corresponding unitary polynomials. The unitary polynomials can be viewed as a set of one-body wave functions of a two-dimensional repulsive Coulomb gas of particles. We then construct the N-body ground state wave functions from these unitary polynomials according to symmetry and statistics requirements. Interestingly, in the case of the zero matrix-model random interaction we precisely obtain the Laughlin wave functions for the fractional quantum Hall effect. In the case of the nonzero matrix-model random interactions, we demonstrate that the N-body wave functions obtained from our normal matrix model describe electrons on a plane under the influence ofnonuniform external electromagnetic fields. Thus, our solutions from the normal matrix model show that fractional quantum Hall effects do exist in complex situations of external electromagnetic fields. It will be interesting to observe such fractional quantum Hall effects experimentally.
Recently, there has been much interest in the relationships between various two-dimensional theories. The rich structures of the ground states in strongly interacting electron systems [1] may be closely related to two-dimensional conformal field theory as well as to the (2 + 1)-dimensional Chern— Simons gauge theory, Abelian and non-Abelian [2]. Another example recently pointed out by Callaway [3] is that the Laughlin wave functions [4] for the fractional quantum Hall effect (FQH) [5] can also be constructed from the point of view of a random matrix model [6]. Matrix models are known to provide a powerful tool for calculating nonperturbative effects in two-dimensional gravity [7,8]. This new connection between the matrix models and the FQH theory is very fascinating, In this paper, we formulate matrix models for the normal matrices and construct the unitary polyno452
mials in the normal matrix models with arbitrary orders of matrix-model random interaction. We cornpare the unitary polynomials to the orthogonal polynomials in the Hermitian matrix models [9] and obtain many identity relations for the unitary polynomials. It is well known that wave functions describing the one-dimensional repulsive Coulomb gas of particles can be constructed from the orthogonal polynomials of the Hermitian matrix models [9]. We show that wavefunctions describing the two-dimensional case can be constructed from the unitary polynomials given here for the normal matrix model. The use of the normal matrix is essential. Since the matrix and its Hermitian conjugate commute, i.e., [M, M~J=0 for a normal matrix M, they can be simultaneously diagonalized. The complex eigenvalues of the nor-
0375-9601/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
Volume 167, number 5,6
PHYSICS LETFERS A
mal matrices can be interpreted as the positions of particles in two dimensions. Taking the unitary polynomials of the normal matrix model as set of one-body ground state wave functions in two dimensions, we can construct the Nbody wave functions ofthe ground state uniquely according to the rules in the matrix model and the U (1) symmetry. In the case of zero matrix-model random interactions, we obtain exactly the Laughlin wave functions for the FQH effects of electrons moving in a two-dimensional plane under the influence of a uniform external magnetic field perpendicular to the plane. This is an interesting coincidence. We next investigate the situation of non-zero matrix-model random interactions. Indeed our exact solutions from the normal matrix model provide new results for the FQH effects. We show that the N-body ground state wave functions obtained describe electron systems on a plane influenced by complicated electromagnetic fields. We demonstrate explicitly that the leading order matrix-model random interactions (g1 0) describe a nonuniform magnetic field perpendicular to the plane and a nonuniform static electric field in the radial direction. We then show the existence of the fractional quantum Hall effects and calculate the filling factor. It will be interesting to observe such effects experimentally. Let M be an Nx N normal matrix, [Mt, M] = 0, with the distinct eigenvalues ;=x1+ iy~where Xj and y~(j= 1, N) are real. Quantum mechanically, our normal matrix models are described by the partition function ...,
Z(g)=
J
~11dM~ exp[ —Tr V(M)],
(1)
where V(M)=IMtM+ n~~ ~ 2N
3 August 1992
to a constant factor, the partition function Z(g) is [61 r N Z(g) j fl dji(~) 4 (z1, ZN) 2, (3) ...,
where du (z~)= d2~d~exp [ V(~)= ~ZJZJ =
—
V(z3)],
g~_1
-
-
n~2 N”’ Z7 Z7
+
I 12 + ~
I I 2n
~
(4)
n~2 N
and 4(z1, 4(z,,
...,
...,
ZN) 15
ZN) =
the Van de Monde determinant
[II
(Z1—~j)
(5)
.
>‘
There is a global U (1) symmetry in eq. (3) under (
,~
~,
~
1
which is equivalent to the symmetry of eq. (1) under M—~UM. From the partition function (3), as done in ref. [9] for the Hermitian matrix models, we can construct a set of polynomials, which we call unitary polynomials. When the dimension of the matrix reduces to one, the partition function represents the probability with the weight exp [ —Tr V(Z)]. This fact means that the unitary polynomials must be of this weight. By defining
r
j fl
~
d~(z,)J4(Z1,
...,
(7)
Z,,) 2,
which is equal to the partition function Z of eq. (3) when n = N, we then construct a set of polynomials of degree n, 2
(2)
and terms with coupling constants g~describe the matrix-model random interactions. The theory is invariant under M-+ UM, where UU~=1, while [M, U] = 0 is required in order that UM is also a normal matrix. Since Mis a normal matrix, there exists a unitary matrix S which can simultaneously diagonalize M and Mt to ZD= (d~~z1) and ZD= (d~j2~) respectively: M= StZDS and Mt = S45t. Then, up
Pfl(Z)=Z;1(g)Jfld4u(ZJ)
X
(z— Z~)
.
I4(Zi,...,z~)I (8)
The leading term in the power of z is Z” with the coefficient unity. Similar to those identities found in the Hermitian matrix models [9] we can obtain the following identities,
453
Volume 167, number 5,6
PHYSICS LETTERS A
n+I
‘
I fl
d~i(~) 4(Z 1,
•‘ j=1
Z~+l)4*(Zl,
...,
...,
2+.... (16) Because the ~ are complex (bnm ~ 0, m < n 1) the P~(Z)=z”+c~,~_2z~ recursion relations among the unitary polynomials
z~)Z~,
—
n +1
I’
=(
JJ d~z(~)4(Z1,...,Z~~,)
1)n+1_k
~ X 4* (Z1,
...,
J
)~
Z~, ...~ ~
(9)
are no longer as simple as those between the orthogonal polynomials in the Hermitian matrix models. For the latter, only three polynomials with orders of n + 1, n and n 1 appear in the recursion relation, which is important in order to derive the string equations by means of the orthogonal poly—
Z~ ~
~
~
~ n+I
j {J dy(~)
=
3 August 1992
4(Z,,
...,
Z~ i)4*(Z,,
...,
Z~)Z~÷,
nomials [7,8]. So the string equations in the normal matrix model has to be derived in a different way Since h~=Z~÷,/(n+l)Z~, Z~can be representedby h~, (17)
Z~=n!h~_,...h,h0,
x ~ (_l)n~_kA*(Zl,...,~k,...,Zfl)z~, (10) k= i 2k
represents the absence of Zk in
where the symbol 4*in eqs. (9) and (10). Itiseasyto seethatthesum in eq. (10) is the complex conjugate of the determinant ~1 1
z, z2
1
...
Z
Zr’
Z~
zr+{
Z~÷,
I
Z?’
which is zero for s= 1,
J
~1)
...,
n
Zn +
1
1. For s = n, we have
—
d/2(Z) P~(Z)P~(Z) hndm,n,
h
=
h~are
(12)
(13)
real, (14)
Zfl+1
(n + 1) Zn Consider the following expansion in the unitary polynomials, —
n+l
ZP~(Z)
=
~
~~P1(Z)
exp[—~V(z)]
,
(18)
Jcb~~ØidZdZ=d/m.
(19)
Z~+j
Hence, the polynomials P~(Z) are complex orthogonal, we call them unitary polynomials,
where
,=
which satisfy the unitary condition
I
Z~ dp(~)P~(z)Z”= n+l~
J
but the relation between Z~and ~ is not as simnomials P in the orthogonal polynomials. The polyple as that 1(z) may be normalized as
b,,,,, Pm (z).
(15)
This concludes our matrix-model formulation for normal matrices. We emphasize that the unitary polynomials constructed are for arbitrary matrix-model random interactions as given in eq. (2). Next we shall calculate N-body wave functions of the normal matrix model using the unitary polynomials derived above in the case of zero matrix-model random interactions; i.e. g~=0and only the Gaussian first term in eq. (2) exists. Very interestingly they precisely give the Laughlin wave function for FQH effects. We investigate the case of non-zero matrixmodel random interactions. We demonstrate that the N-body wave functions obtained from our normal matrix model describe electrons on a plane under the influence of nonuniform external electromagnetic fields, for which we also show the FQH effects exist and calculate the filling number. Thus the normal matrix model provides new insights for FQH effects
m=O 1*1
Making use of eq. (13), we have b~~÷1 = 1 and = 0, i.e. the unitary polynomials have the form 454
For a general complex matrix model, the string equation has been discussed in ref. [10], which contains the string equation for the normal matrix model discussed here.
Volume 167, number 5,6
PHYSICS LETTERS A
from which we illustrate that fractional quantum Hall effects do exist in quite complex situations of external electromagnetic fields. Let us now briefly describe the physical system of the FQH. In the FQH, the N-electron system is restricted to an (x, y)-plane and the spin of the electrons is frozen by a uniform external magnetic field, B, perpendicular to the plane. The one-electron Hamiltonian in the FQH is H,=_~i__(~V+~A”1, 2me\i C J
(
1k Z ~t
1,
...,
Zn)
N
=
fl
/ (~,—~)~“~‘
exp( \
1>J
N —~
itary polynomials 0, of eq. (18) are the one-body ground state wave functions for the one-electron Hamiltonian H1, H~ Oi = I hw~0,.
~ 1Z112)
(21)
‘=‘
for the Hamiltonian where the background and the Coulomb interactions u between N electrons are small, ~ H~(~) + u.
(22)
(24)
Since the set of On is complete, the following functions may be expanded in 0,, 0~:::~,raP~(Z)...P~,(z)exp[—IVo(Z)1
(20)
where A is the electromagnetic potential of an external magnetic field B. In order to write down the Schrödinger equation explicitly, usually the symmetric gauge, A= ~B(y~—x9) is used. The important result [4,5] is that the variational ground state wave functions of N-body electrons are
3 August 1992
=~a,Ø,,
(25)
where a...b and n...m are arbitrary positive integers and the a, are complex constants. It is easy to check that O~:::~ are also the one-body ground state wave functions. From On we then construct the N-body ground state wave functions, eq. (21), from the point of view of the normal matrix model. Note that the partition function, eq. (3), for g~= 0 is exactly the probability associated with the Laughlin wave function Y’k=~o. Then,becausethereisthenormoftheVandeMonde determinant in the measure of the integration, all Nbody wave functions in the frame ofthe matrix model must carry the determinant factor. The U (1) symmetry of the theory requires that the N-body wave functions are U (1) covariant, which implies homogeneity of these functions in the z3. This requirement coincides nicely with the requirement of definite angular momentum for the N-particle system,
Therefore Hwk(zI,
...,
ZN)=
[INhwC+O(u)]~fk(Zl,
...,
ZN),
i.e. the wave function is an eigenfunction for the angular-momentum operator
(23) .whére the magnetic has been taken uni2=l,length and wc=eB/mec is to thebecycloty, l=(hc/eB)” tron frequency. The ground state energy is lifted by 0(u) ~ In order to use the normal matrix models to discuss the FQH, we identify the matrix M of our matrix model as the position operator for the N electrons and the complex eigenvalues of M as the twodimensional positions ofthe electrons. Thus it is natural to use normal matrices so that the position operator matrix Mand its Hermitian conjugate Mt can be diagonalizable simultaneously by a unitary matrix. We then observe that for g~=0, V(Z)— V 2), the normalized set of the un0(Z) ~exp(—~ 1Z1 .~
N ~
(~ ~
— ~
Moreover, since we are considering an N-fermion system, we require the N-body wave functions cornposed from the one-body wave functions, after factoring out the Van de Monde determinant that is totally anti-symmetric in (Z~, ZN), to be symmetric. The unique N-body ground state wave functions satisfying the above requirements are ...,
455
Volume 167, number 5,6
PHYSICS LETTERS A
Nmec
2k
~
where ~ is the unit radial vector in the (x, y)-plane and ~ is perpendicular to the plane. The vector po-
e’J’Po(ZI)PI(zi)...PN_l(Zl)) I
x exp(
—
I
~
V0
(~5
))~
(26)
where ~ is + 1 for even permutations of 1, N and —1 for odd ones. The sum of i,j, /in eq. (26) is nothing but the determinant ‘~“
...,
tential A and the scalar potential ~‘ in the symmetric gauge are A = IB0( 1+2 ~
...,
det[Pfl_l(~)]=det(z7’)=A(ZI,...,ZN). (27) The wave function (26) changes covariantly by exp[~iN(N—l)(2k+l)O]whenthez 3transformas given by eq. (6). Equation (26) is precisely the Laughlin wave function Wk, eq. (21), with arbitrary k. When there is a quasiparticle excitation in the system, the Laughlin treatment [4,5] for this can be used. First, a factor Z Z0 where Z0 is the position of the quasiparticle, is multiplied with 0~:::~ of eq. (25), except 02:::~.Because of the completeness of the set of 0’s, the product is also a one-body ground state wave function. Then, according to our construction, the resulting N-body wave functions are —
N j=’
x exp(
—
2”~’
U (~—Zo)[4(Zl,...,zN)]
Wk(Zl,,ZnZO)
~
(30)
Nmec The Schrödinger equation for the electron is ~°
H~(g~)~v=Eyi, where H, (g1) =
~
~-~—
(3la)
(~
2
0~+ A.,) ~
—
eq,
(31 b)
j=x,y i
with A1 and ~ given by eq. (30). After some calculations one can show that the ground state wave function of eq. (31) is just ~, =
I
p, exp [ — V( Z, g,)]
,
(32a)
where V(Z, g,) and is V(z) of eq. (4) with g~~tO and g~=0forn>l (32b)
(Z5))~
(28)
which is U (1) covariant, corresponding to the coyariance of the determinant det(M—Z01) under M-+ UM. This wave function describes the quasiparticle excitation in the system with the charge e/ (2k+ 1) [11]. After establishing that in the case of zero matrixmodel random interactions the unitary polynomials of the normal matrix model can give Laughlin’s wave function, we are interested in finding out what new physical phenomena our normal matrix model describes in the cases of non-zero matrix-model random interactions, Consider the case of an electron system confined to the (x, y)-plane under the influence of the fields B(r) = _B
456
r2) ~
4g hB0 r
V 0
0(l +
(29)
E(r)= S~ihBo~p
Wk(ZI,...,ZN)=A(ZI,...,ZN)
I,)
3 August 1992
~-
r2)z,
Thus the nonvanishing leading order matrix-model random interaction, g, ~ 0, describes influences of nonuniform electromagnetic fields of eq. (29) on charged particles in the (x, y)-plane. Next we can construct the N-body wave functions for this g, ~ 0 case similar to what we did in the case of zero matrix-model random interaction. The variational ground state for the N-body Hamiltonian N
H(g, ) = ~ H, (z.,, g, ) + u
(33)
i=’
is taken to be ~vk(Zj, ZN, g,) as given by eq. (26) except that V0 is replaced by V(g1). The square of the modulus of ~‘~(g,) can be written as a classical Boltzmann distribution 2=expE—fl~eff(Z1,...,zN)], (34) ...,
IW(Zl,...,ZN)I
where l/fl=2k+ 1 is the fictitious temperature and
Volume 167, number 5,6
PHYSICS LETTERS A
3 August 1992
2
the effective potential cPeff=2(2k+l)2 ~ ln~z
a~~_O, axY=2k+lh. 1 e
(40)
1—~
I
2+
4).
+ (2k+ 1) ~ (i.~1 This effective potential describes the following physical situation: the first term in eq. (35) represents the repulsion between particles of charge 2k+ 1 via the two-dimensional Coulomb interaction; the second term is the attraction of these charges to the or2/ igin due to a charge density Po (1 /27t) (1 + 8g,r N), i.e., = —2 (2k+ 1)2
+2(2k+ 1)
~
k
J
~ ln I Z
—
‘<3
dZ2
Po( IZI
I
(k)
Po = 2k+l 1 eB(r) 2k+l hc
— —
Pe
bitrary interactions. We cannormal interpretmatrix-model these unitaryrandom polynomials as a set
) lnIZ—Zk I
Thus, the electron density in state
~1Ik(g,)
.
(36)
is equal to (37)
where B( r) is the value of B( r) ofeq. (29). In other words, the state wk(~I) has the filling factor v = (1 + 8g 2/N) / (2k+ 1), which has a variation in 1r the radial direction. To see the physical implication corresponding to such an electron density and filling factor, we apply a uniform electric field E 0 in the (x,y)-plane and analyze the situation classically. Using complex variables for vectors in the plane and setting z= X+ ~Y, v= E, =E0 + E( r) =E~+iE~,the Newton—Lorentz equation for the electron is ~,
eE~—La) m
— —
+
~
(38)
Solving the equation, the electron drift velocity arising from the electromagnetic field is eE, 2/N) +0(E?). (39) imw~(l+8g,r In the linear approximation eq. (39) is consistent with the relaxation-time approximation in transport theory. Since the electric currentj= —peevd, using eq. (37) and considering the linear approximation, we have Vd =
—
This impliesmagnetic that the FQH exist when the perpendicular field effects and the applied electric field are not uniform. This is a new physical conclusion provided by our normal matrix model with g 1 #0 and g,, ~ 2 = 0. (More complicated physical situations are implied by the general gn #0 cases.) This implies that FQH effects can exist in rather complicated situations of external electromagnetic fields. It is important to verify this experimentally. In conclusion, we conclusion have constructed the normal matrix models and their unitary polynomials for ar-
of one-body wave functions of a two-dimensional repulsive Coulomb gas. We can then also construct an N-body ground state wave function. For vanishing matrix-model random interactions we obtain Laughlin’s for theonfractional quantum Hall effect wave wherefunction the electrons the two-plane are under the influence of a uniform magnetic field perpendicular to the plane. For nonvanishing matrixmodel random interactions, the normal matrix models provide a useful way to construct N-body ground state wave functions associated with more complicated electromagnetic fields. We explicitly demonstrate that the first term of the matrix-model random interactions (i.e., the g~#0 term) corresponds to the situations of a nonuniform external magnetic field perpendicularto the plane and a static electric field in the plane. We also show that the fractional quantum Hall effect exists in these complicated electromagnetic fields. We also can calculate the filling factor. It will be interesting to observe these effects experimentally. This work was supported in part by the US Department of Energy (DOE) and INCOR of the University of California. Y. Yu was also supported by the Foundation of National Natural Science of China and the Foundation for Young Researchers in the Academia Sinica and the Grant LWTZ-1298 of the Chinese Academy of Sciences. He would like to thank the Physics Department of the University of California at Davis for the warm hospitality.
457
Volume 167, number 5,6
PHYSICS LETTERS A
References [1] F.D.M. Haldane, Phys. Rev. Lett. 51(1983) 605; B. Halperin, Phys. Rev. Lett. 52 (1984) 1583; X.G. Wen, Int. J. Mod. Phys. B 2 (1990) 239; Phys. Rev, B 40 (1989) 7387; X.G.Wen and Q. Niu, Phys. Rev. B 41(1990) 9377; B. Blok and X.G. Wen, Princeton preprint IASSNS-HEP90/23 and 55, to appear in Phys. Rev. B; LASSNS-HEP- 90/ 66; N. Read, Phys. Rev. Lett. 65 (1990) 1502. [2] R. Jackiw, Phys. Rev. D 29 (1984) 3975; J. Frohlich and T. Kerler, Zurich preprint ETH-TH/90-26; Nucl. Phys. B 354 (1991) 369, and references therein; X.G. Wen, Phys. Rev. Lett. 66 (1991) 802; Princeton preprintlASSNS-HEP-91/20; J. Fröhlich and A. Zee, Nucl. Phys. B 364 (1991) 517; A. Zee, preprint NSF-ITP-9 1-41. [3]D.J.E.Callaway,Phys.Rev.B43(l991)8641. [4] RB. Laughlin, Phys. Rev. Lett. 53 (1983) 722. [5] R.E. Prange and SM. Girvin, eds., The quantum Hall effect (Springer, Berlin, 1988). [6] M. Metha, Random matrices and the statistical theory of energy levels (Academic Press, New York, 1967).
458
3 August 1992
[7] V.A. Kazakov, Phys. Lett. B 150 (1985) 282; F. David, Nucl. Phys. B 257 (1985) 45; V.A. Kazakov, I. Kostov and A. Migdal, Phys. Lett. B 157 (1985) 295; D. Weingarten, Phys. Lett. 890 (1980) 280. [8] E. Brézin and V.A. Kazakov, Phys. Lett. B 236 (1990) 114; M.R. Douglas, Phys. Lett. B 238 (1990)176; MR. Douglas and S. Shenker, Nucl. Phys. B 335 (1990) 635; D.J. Gross and A. Migdal, Phys. Rev. Lett. 64 (1990) 127;
Nuci. Phys. B 340 (1990) 333; T. Banks, M.R. Douglas, N. Seiberg and S. Shenker, Phys. Lett. B 238 (1990) 279. [9] E. Br&in, C. Itzykson, G. Parisi and J.B. Zuber, Commun. Math. Phys. 59 (1978) 51; C. ItzyksonandJ.B. Zuber,J. Math. Phys. 21(1980)411; D. Bessis, Commun. Math. Phys. 69 (1979) 147. [10] J. Ambjørn, J. Jurkiewicz and Yu.M. Makeenko, Phys. Lett. B251 (1990) 517. [11] RB. Laughlin, in: Springer series in solid state science, Vol. 53, eds. G. Bauer, F. Kuchar and H. Heinrich (Springer, Berlin, 1984).