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Hilbert space for charged particles in perpendicular magnetic fields
ANNALS
OF PHYSICS
215, 233-263 (1992)
Hilbert Space for Charged Particles in Perpendicular Magnetic Fields* GERALD Department Massachusetts
V. DUNNE
of Mathematics and Center ji)r Theoretical Physics, Institute of Technology, Cambridge, Massachusetts 02139 Received October 9. 1991
We describe the quantum mechanics of two-dimensional charged particles in a perpendicular magnetic field in the planar Landau and spherical monopole configurations. These models, particularly in the work of Laughlin and Haldane, are crucial to the theoretical understanding of the quantum Hall effect. Here we present the full Hilbert space structure in each case, with special emphasis on the relationship between the two systems. The formulation in terms of stereographically projected complex coordinates makes the connection especially explicit and naturally generalizes to more complicated two-dimensional surfaces where the interaction of the particles with an external perpendicular magnetic field may be regarded as an interaction of the particles with the two-dimensional (Kahler) metric of the surface. This generalization is illustrated by the hyperbolic configuration of particles constrained to the upper sheet of a hyperboloid in the presence of a hyperbolic monopole. ‘c 1992 Academic Press. Inc.
I. INTRODUCTION
Great advances (both experimental and theoretical) have been made in recent years in the understanding of two-dimensional systems of non-relativistic charged particles. One such advance, the quantum Hall effect, involves a two-dimensional system of electrons in a perpendicular external magnetic field. The quantum behavior of such a system, especially at very low temperature and very high magnetic field strength, holds many interesting surprises [l]. A major theoretical step was taken with Laughlin’s proposal [2] of variational wavefunctions describing incompressible states corresponding to special rational filling fractions of the highly degenerate lowest Landau level. Haldane [3] introduced a variant of the Laughlin wavefunctions by considering a system of electrons constrained to the surface of a sphere with a magnetic monopole at the center of the sphere. The singleparticle states for such a system were originally studied by Dirac [4], Tamm [S], and Fierz [6] and are essentially the “monopole harmonics” of Wu and Yang [7]. A significant advantage of the spherical geometry system is that it permits transla* This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under Contract DE-AC02-76ER03069. and NSF Grant 87-08447.
233 0003-4916/92 $9.00 CopyrIght ,(‘I 1992 by Academic Press. Inc. All rights of reproducfmn m any form reserved.
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tionally and rotationally invariant states with finite particle number and is especially suited to numerical simulations [3, 81. The finite area of the spherical surface translates into finite degeneracies of the energy levels, and this allows for a simpler treatment of the thermodynamic limit. In this paper we describe the full Hilbert space structure of the planar Landau system [9] and of the spherical monopole system, and we describe precisely the relationship between the two models. The monopole system is a consistent deformation of the planar system, with the deformation parameter being the inverse of the monopole charge g. In the limit g + cc all features of the Landau model are regained from the corresponding features of the monopole model. The magnetic field strength at the spherical surface (of radius R) due to a monopole of charge g is B = g/R2, so the limit g + co corresponds to the infinite radius limit in which B is kept fixed. Previous discussions of the Landau [lo] and monopole [3,8, 1 l] systems in the context of the quantum Hall effect have concentrated on the lowest Landau level-here we incorporate all excited states into the Hilbert space. This is motivated by the need to study interactions between the electrons (especially at higher temperatures and in weaker B fields) and also by the obvious desire for theoretical completeness. We show that by a suitable choice of stereographically projected coordinates the relationship between the Landau and monopole systems becomes explicit and transparent. The full Hilbert space in each case may be found either by solving a second-order linear ordinary differential equation or by making use of related algebraic structures to reduce the problem to a first-order linear differential condition. In the planar case the algebraic structure arises from the Heisenberg algebra of non-commuting magnetic translations [9], while the relevant algebraic structure in the spherical case in SU(2) [S-7]. The wavefunctions, in each case, can be expressed as matrix elements of the coherent state operator for the corresponding algebra. We also show that these two models are special cases of the more general model of charged particles constrained to the surface of a two-dimensional Kahler manifold, interacting with an external gauge field which has its field strength two-form proportional to the surface’s Klhler form (this says that the particles move in a uniform magnetic field per unit area which is everywhere normal to the surface). Another example of this structure is that of particles constrained to a hyperboloid in the presence of a hyperbolic monopole. This paper is organized as follows. In Section II.A, the full Hilbert space is described for the Landau system of planar charged particles in the presence of a constant normal magnetic field. In Section 1I.B we discuss the algebraic step operator approach to this and show that the wavefunctions are matrix elements of the coherent state operator for the Heisenberg algebra. The Hilbert space structure of the spherical monopole system is derived in Section IIIA, and the corresponding algebraic analysis is given in Section 1II.B. Here the wavefunctions are matrix elements of the SU(2) coherent state operator. In Section IV we discuss a synthesis of these models in terms of two-dimensional Kahler manifolds and illustrate this synthesis with another related model, that of the hyperbolic monopole. Here the relevant algebraic structure is the (non-compact) Lie algebra of SU(1, 1). In this
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paper we restrict our attention to the discrete part of the spectrum in the hyperbolic system. For a complete discussion of this model, including the continuous part of the spectrum, see Refs. 26 and 27. An Appendix contains some properties of the Laguerre polynomials, Jacobi polynomials and associated Jacobi functions.
II. LANDAU
SYSTEM
A. Hilbert Space In this section we review the Landau problem [9] of non-interacting charged particles moving in the two-dimensional plane in the presence of a fixed external magnetic field (of strength B> 0) perpendicular to the plane. The quantum mechanical Hamiltonian operator for this system is (we work in units ti=m=e=l)
=$(-i8-A)2,
(2.1)
where we choose the vector potential A in the symmetric gauge
It proves more convenient to work in terms of the resealed complex coordinates z=$iji(xl
+ix2)
‘1 In these coordinates the Landau Hamiltonian
(2.1) is
H,=B(-~~-&-~~)+~z~),
where 8 = a/&, 8 = a/&. The planar angular momentum J=zd-Z(q
(2.4) operator (2.5)
commutes with H,. We shall therefore seek simultaneous eigenstates II/ of energy and angular momentum. To define the single-particle Hilbert space, we extract an exponential measure factor $(z, 5) = e-(1P) l=l*$(z, 2).
(2.6)
The “reduced” wavefunctions $ will be identified with elements of the Hilbert space,
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while the exponential factor contributes to the measure in the Hilbert space inner product: (2.7a)
In (2.7), f and g are functions of both z and Z. The bar ~ over f in (2.7) denotes complex conjugation, as in (2.3). The inner product (2.7) is a generalization of that considered by Bargmann[ 121 in his description of the Hilbert space of analytic functions on the plane, and which has been applied to the lowest Landau level of the Landau system by Girvin and Jach [lo]. Note, however, that in order to incorporate the higher Landau levels we cannot restrict the functionsfand g to be holomorphic-the appropriate generalization is described below. Having identified the “reduced” wavefunctions I,& in (2.6) with elements of the Hilbert space, one should identify the operators acting in the Hilbert space as those acting on 6, rather than on $.’ The relation between an operator 0 acting on II/ and the corresponding Hilbert space operator 6 acting on I,& is simply one of conjugation d=e
(l/2)
Irl’o,-(l/2)
/=I?.
(2.8)
Thus, the Hilbert space energy and angular momentum
operators are
A,=B(-a8+2++),
(2.9)
.Lza-zLi
(2.10)
The lowest Landau level, of energy B/2, is described by holomorphic wavefunctions lj = l)(z). Now write an arbitrary wavefunction $ as an angular momentum eigenstate $=z-‘P(lz12),
(2.11)
where P is some function to be determined and j is an integer to ensure singlevaluedness of 6. Requiring $ to also be an energy eigenstate, @,$=E$
(2.12)
then leads to the following ordinary differential equation for the function P(x): x$+(j+l-x)$+ ’ As noted by Girvin and Jach [lo] in the context [ 131 procedure of not having the differential operator of the form $ in (2.6).
(
5-t
>
P=O.
(2.13)
of the lowest Landau level, this explains Laughlin’s 8 act on the exponential part of the wavefunctions
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This is a differential equation of Laguerre form, xh” + (j + 1 -x)/r’
+ nh = 0
(2.14)
which has as regular solution the generalized Laguerre polynomial h = L’,(x)
(2.15) Further properties of the Laguerre polynomials are listed in the Appendix. Comparing (2.13) and (2.14) we obtain the solution
$~=&&z-j$z’L’,(Iz12), E=B(n+$),
(2.16)
n = 0, 1) 2, ...) j= --n, --n + 1, ... . The normalization
factor arises from the Laguerre integral [14] ccd.xe-“x’L.;(x)
(n+j)! L’,(x) = 6,, ,r.
(2.17)
.r 0
Note that Lx is defined (and normalizable in the sense of (2.17) being finite) for j>, -n, and when the upper index is negative one may also write (2.18) From (2.16) we recognize n as the familiar “Landau level” index [9] and we see that the Hilbert space 2 decomposes into an infinite number of such Landau levels
each Xn being infinitely degenerate-the degenerate states being distinguished by their angular momentum index j. Note that the lowest Landau level states (i.e., those with it = 0) all have non-negative angular momentum, while the higher Landau levels, Xn with n 2 1, contain states with negative angular momentum. It follows from (2.17) that the wavefunctions 1+6i,in (2.16) are orthonormal with respect to the Hilbert space inner product (2.7). In fact, they also form a complete
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orthonormal set of wavefunctions [15, 161. We can therefore express any element of the single-particle Hilbert space as a linear superposition (2.20)
where the numerical coefficients a,, j must be such that $ is normalizable with respect to (2.7). In certain physical applications (for example, when B is very large [ 171) it is instructive to consider just the lowest Landau level Z. portion of the Hilbert space. As mentioned previously, the above discussion then reduces to Bargmann’s Hilbert space of analytic functions with basis functions I)~=~=-$z~,
j=O, 1,2, ....
(2.21)
which form a complete orthonormal set for analytic functions [12]. (Note that the wavefunctions $i, in (2.16) form a complete orthonormal set for functions of z and 5.) Within & a special role is played by Bargmann’s “principal vectors” exp(wz)
(UEC).
(2.22)
These functions play a role similar to that of the Dirac b-function in the conventional position space representation of quantum mechanics. They arise as a resolution of the identity within the restricted Hilbert space Zo, f
tj’,=,(o)
g’,=,(z) = exp(&),
(2.23)
j=O
and have the important
property
f(u) =f(z), 5b(w) exp(6z)
(2.24)
which illustrates their similarity with the Dirac a-function. B. Algebraic Approach The Hilbert space structure described thus far has a simple algebraic formulation which is of great practical use in explicit computations. (It also provides the key to a systematic analysis of the many-body quantum mechanics of anyons in a magnetic field [18].) The idea is very simple-the energy and angular momentum operators may be expressed in terms of two (commuting) sets of harmonic oscillator step operators, H, = B(ata + i)
(2.25)
J= btb - ata,
(2.26)
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where
(2.27)
Then the a’s and b’s commute with one another, but amongst themselves they satisfy the Heisenberg algebra commutation relations
[a,a+]= n,
[b,b’]=Q.
(2.28)
From (2.25)-(2.28) we see that bt and b have a simple interpretation as raising and lowering operators (respectively) for angular momentum. As they commute with H,, they act within a given Landau level connecting states of equal energy. Similarly, the ut and a act as raising and lowering operators (respectively) for energy, while simultaneously acting as lowering and raising operators (respectively) for angular momentum. They therefore connect states in different Landau levels. The operators atbt and ab act as raising and lowering operators (respectively) for energy, while leaving the angular momentum unchanged. To consider their action on the wavefunctions $ we should examine the conjugated operators Lit, 4, @, 6 defined as in (2.8), $=a-=
(j= -a p=
-$+z
(2.29)
6= a. These operators, of course, satisfy the same commutation relations as the corresponding a’s and b’s, and the relations (2.25) and (2.26) simply acquire “hats” (“““) throughout. Moreover, with respect to the inner product (2.7) it is easy to verify that ri and tit (and also 6 and !?) are truly adjoints, i.e., (2.30)
The action of these Hilbert space operators on the wavefunctions $j, in (2.16) is (2.31a) (2.31b) (2.31~) (2.31d)
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In verifying (2.31a)-(2.31d) we use the differential-difference and difference relations for the Laguerre polynomials listed in the Appendix. Note also that the “bottom ” of the nth Landau level is the angular momentum state with j= --n because &$L= M-n= 0. The n th Landau level is unbounded above, since $i, is never annihilated by 6’. We conclude this discussion of the algebraic structure of the Landau system’s Hilbert space by showing that the wavefunctions 4; have a simple representation in terms of matrix elements of coherent states for the Heisenberg-Weyl group. Define a “vacuum” state IO) such that b(O)=0
(2.32)
and excited states by (2.33)
One can then define the coherent state 1~) by [19]
lz> = Wz) IO>,
(2.34)
where D(z) is the coherent state operator D(z) = exp(zb+ - Fb) :b+ -Z/I =e -(I/Z) (iI ee.
It is then straightforward (n+jl
(2.35)
to compute that [20] D(z) In) =~~e-1’2”‘2~iLi,(lzJ2) =e
-(l/2)
[=I2 “j $ nr
(2.36)
which are the Landau wavefunctions, including the exponential measure factor. Various authors [ 10, 211 have noted this correspondence between coherent states and the Landau wavefunction in the lowest Landau level (n = 0), where they argue that the two-dimensional configuration space behaves as a phase space due to the very strong B field which energetically isolates the lowest Landau level. In fact, the correspondence to coherent states generalizes very simply to higher Landau levels, as shown by (2.36). III. MONOPOLE SYSTEM A. Hiibert Space In this section we analyze the quantum mechanics of charged particles constrained to the surface of a sphere with a magnetic monopole at the center of the
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sphere. This problem was originally considered by Dirac [4], Tamm [S], and Fierz [6] and has recently been revived in the context of the fractional quantum Hall effect by Haldane [3]. Here, we reformulate the system in a form which permits an immediate and simple comparison with the Landau model discussed in Section II. The qualitatively analogous features of the two systems are clear--each refers to quantum mechanical charged particles moving on a two-dimensional surface in the presence of a uniform, everywhere perpendicular magnetic field. Intuitively, one would expect that as the sphere’s radius becomes very large the spherical surface becomes “essentially flat” so that the monopole system reduces to the Landau model. The purpose of this section is to make this intuitive expectation precise. The monopole system is formulated as follows. Consider the two-dimensional spherical surface to be embedded in three-dimensional Euclidean space, for which we use both Cartesian (x’, x2, x3) and spherical (Y, 8, 4) coordinates x1 = r sin 19cos 4 x2 = r sin 6 sin Q
(3.1)
x3 = r cos 8.
We represent a magnetic monopole of (positive) charge g, centered at the origin in three-space, by a vector potential expressed in the “Dirac string” gauge,
A=( -$p,&q,~).
(3.2)
The components of the vector potential along the polar unit vectors i, 4, and 4 are A,=A,=O (3.3) A,= --&(1-c0”0).
Note that this vector potential has a singularity along the negative x3-axis-the significance of this will be discussed later in relation with the planar Landau model. The magnetic field corresponding to the vector potential in (3.2) (or (3.3)) is directed radially outwards from the origin: B=VxA
The Hamiltonian governing the motion of charged particles in three-space in the presence of the monopole is
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(3.5)
Restricting the motion to the surface of the sphere r = R, we obtain the required two-dimensional monopole Hamiltonian fz,++ia-A))‘,,=, 1
-- i
a
2R2 sin 6 80 sin
a e-86
1
a?
2R2 sin’ 0 @ (3.6)
At the surface of the sphere the magnetic field is everywhere perpendicular surface, and of constant magnitude: B = g/R’.
The Dirac quantization g to satisfy
condition
to the
(3.7)
[4] requires the dimensionless monopole charge 2g = integer.
(3.8)
The key to understanding the relationship between the monopole system and the Landau system is to transform to stereographically projected coordinates which project from the “South pole” (0, 0, -R) of the sphere to the tangent plane to the sphere at the “North pole” (0, 0, Rksee Fig. 1. In such a projection, the South pole itself is mapped to the point at infinity in the projected plane, so it is in fact more appropriate to talk of the monopole system with Hamiltonian (3.6) defined on the spherical surface r = R with the point at the South pole excluded. As we shall see, one coordinate patch, with the South pole excluded, is sufficient for the purpose of comparing with the Landau system. This explains why it is sufficient to use the vector potential A in the Dirac string gauge (3.2)-to consider the model on the entire spherical surface would require (at least) two patches, a system for which the Wu-Yang [ 71 formulation is more appropriate. The aforementioned stereographic projection is achieved by the coordinate transformation
(3.9)
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N-lz+---
P’
S
FIG. 1. A cross section, at fixed polar angle 4, of the stereographic projection of the point P on the surface of the sphere of radius R, from the South pole S to the point P’ lying in the tangent plane to the sphere at the North pole N. N is mapped to the origin of the tangent plane and S is mapped to the point at infinity. The distance (11 is (z( = 2R tan(0/2).
The extra resealing factor JB/2 has been introduced to conform with the normalization of the complex coordinates (2.3) used in Section II for the Landau system. In terms of these coordinates, the monopole Hamiltonian (3.6) becomes
H,=B _ *+Iz(z2a&’ I+!?!! (z&q++2 (LJ
2LJ
).
(3.10)
Now consider the limit in which the spherical radius R + cc. From (3.7) it is clear that in order to maintain a constant magnetic field B one must simultaneously perform the limit g -+ co (in such a way that B = g/R2 remains fixed). Therefore, the scale R may be eliminated in favor of B, the constant dimensionful quantity, in which case the desired limit is simply that of letting the dimensionless charge g -+ co. In this limit, it now follows from (3.10) that H,-tB(-a~-1(za-i~)+flz12)
(3.11)
which is precisely the Landau Hamiltonian (2.4). To determine the Hilbert space of eigenstates for the monopole Hamiltonian (3.10) we proceed exactly as for the Landau model in Section II. First extract a measure factor from the wavefunctions (at this point, this may be viewed simply as a convenient transformation to simplify the eigenvalue problem) by defining 1 bhz, -3 = (1 + JzI2/Zg)” ij(z, 2).
(3.12 )
1 - lzl2/2 ,‘EZ (1+ ,Z,2,2g)g=e ’
(3.13
Note that
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which is the same measure factor as was extracted from the Landau wavefunctions-see (2.6). Once again, the “reduced” wavefunctions 4 will be identified with elements of the Hilbert space, and the other factor in (3.12) contributes to the measure in the Hilbert space inner product: (3.14a) dz d.5 2g+l d/i(z) = ___ 4ing (1 + 1z/2/2g)28+2’
Then the reduced Hamiltonian wavefunctions $ is
(3.14b)
operator which acts directly on the Hilbert
space
(3.15)
whose g + cc limit agrees with I?, in the Landau model-see (2.9). The operator j= zd -ia commutes with I?, and generates rotations about the origin in the projected plane (which correspond to rotations about the x3-axis in terms of the original sphere). Since j and I?, commute, we seek simultaneous eigenstates-we therefore decompose $ as in (2.11). However, it is now more convenient to express the undetermined function P as a function of (3.16) rather than simply as a function of (z(*. Thus we write (3.17) The energy eigenstate condition fi, I,$= E$ leads to the following ordinary differential equation for the function P(x): (3.18)
(‘--~‘)~+2(g-~-(g+l)x)~+2g(~-~)p~~. This is a differential equation of the Jacobi form (1 -x2)h0+(p-cr-(M+p+2)x)h’+n(n+a+/?+
l)h=O
(3.19)
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whose regular solution in the interval [ - 1, l] is the Jacobi polynomial h = P;,p’(x)
-=%(l
-x-3
(I-J)‘l$
=~~,(“;“)(~‘3
{(l -X)X+‘r (1 +.#+n) (x- l)‘lP,
(3.20)
(s+ 1)“.
Further properties of the Jacobi polynomials are listed in the Appendix. Comparing (3.18) and (3.19) we obtain the wavefunctions n! (2g+2n+
g$:, =
l)I+z+2g+ 1) 1) T(n+2g-j+
(2g+ 1)(2g)‘T(n+j+
Z,p;%-jj
(i + ~~~~~~),
1)
E=B n+l+n(n+l) ( 2 37 1 n=o,
(3.21)
1,2, ...
j= -n, -n+ The normalization
1, ...) n+2g. factor arises from the Jacobi integral [22]
1 I -I
dx( 1 - X)%(1 + x)b Pa-P) n (x) Fyyx) = s,,
2”f”T(n+cr+l)z+z+fl+l) n! (2n+a+B+ l)z(n+a+~+
1)’
(3.22)
Note also that the Jacobi polynomial Pjla,p)is defined for a >/ -n, /I b - n: this leads to the upper and lower bounds on the allowed values ofj in (3.21). From (3.22) we see that the wavefunctions @jn in (3.21) form an orthonormal set with respect to the inner product (3.14). In fact, this set is also complete [16]. Let us first comment on the energy and angular momentum spectrum in (3.21). There is still a “Landau level” structure in the sense that the non-negative integer n determines the energy. However, the dependence of the energy on n is no longer linear-it has a quadratic part also. But as g + co the quadratic part vanishes leaving an energy spectrum of Landau form. Each “Landau level” (i.e., the set of states all having the same index n) is once again degenerate, with degenerate states being distinguished by their angular momentum index j. However, the degeneracy of each level is now finite, there being an upper bound n + 2g for the angular momentum in the nth “Landau level”: degeneracy(nth level) = 2g + 2n + 1.
(3.23 )
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In the limit g + co the degeneracy becomes infinite and the degeneracy per unit area degeneracy(nth level) 2g + 2n + 1 = area 47cR2
(3.24) which is the familiar Landau result [9]. To understand the behavior of the wavefunctions 5,&j, themselves in the limit g + co, we use the limiting relation between Jacobi and Laguerre polynomials c141, lim Pjl’,@) 1-g = L:(X). 8-m ( P1
(3.25)
Therefore, (3.26) and we see that the monopole wavefunctions (3.21), indeed, reduce to the Landau wavefunctions (2.16) in the limit g+ co. As mentioned in Section II, it is sometimes possible to restrict one’s attention to just the lowest “Landau level,” n = 0. Then there are 2g + 1 states spanning this level
g$;=o= (2g)!/(2g)‘j! The corresponding
(2g-j)!
zi 2
j = 0, .... 2g.
resolution of the identity in the lowest “Landau
=I,;.
(
- 2g >
(3.27) level” is
(3.28)
In the limit g -+ cc, this reduces to exp(zti), agreeing once again with the corresponding Landau result (2.23). Furthermore, using the orthonormality relations
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(3.22), it follows that this resolution of the identity within the restricted (n = 0) Hilbert space, Sdr(w)(l
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acts like a Dirac d-function
(3.29)
+~)2gf(o)=tli),
for any holomorphic function f(z). To conclude this section we note the wavefunctions in (3.21) are precisely the Wu-Yang monopole harmonics [7] re-expressed in terms of the projected coordinates for the coordinate patch covering the spherical surface minus the South pole. Excluding the South pole in the monopole system is natural for comparison with the Landau model, as the South pole corresponds to the planar point at infinity. B. Algebraic Approach Just as in the Landau system, the monopole system has an algebraic formulation which considerably simplifies the spectral problem, The key fact [S-7] is that the Hamiltonian (3.10) may be written (up to an additive constant) as the quadratic Casimir operator for the Lie algebra N(2). One then finds that the nth “Landau level” (of degeneracy 2g + 2n + 1) corresponds to a finite-dimensional spin (g + n) representation of SU(2). All aspects of the Hilbert space may be recast in terms of the representation theory of SU(2). This becomes especially clear in the stereographically projected coordinates. The relation to the Landau problem may be understood as a contraction [23] of the Lie algebra of SU(2) to the Heisenberg algebra. Define the differential operators, (3.30a) (3.30b) (3.3Oc)
These are generators of the Lie algebra of SU(2), in Chevalley form, CL,, L-1 =2L,
C-LL*l= Then the monopole Hamiltonian
(3.31)
s,.
(3.10) may be written as ;(L+Ld+LpL+)+L;-g2
> (3.32)
595!215,2-2
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where C, is the quadratic Casimir operator for SU(2). Also, the angular momentum operator, J, is related to L, by J=L,+g.
(3.33)
From (3.32)-(3.33) it follows that the problem of finding simultaneous eigenstates of J and H, is essentially the same as simultaneously diagonalizing L, and the quadratic Casimir operator C2 for X42). The operators L, act as raising and lowering operators for the eigenvalue of I,-hence they act as raising and lowering operators for angular momentum. They are thus the analogues of the raising and lowering operators ht and b (2.27) in the Landau system. Having expressed the Hamiltonian as in (3.32) it is now trivial to generate the entire spectrum of states. H, may be rewritten H,=;(L+Lp
+L,(L,-
l)-g2).
Expressing a wavefunction + as an eigenstate of angular momentum J (and hence also of L3) we see that we simply have to diagonalize L, Lp . This is achieved by first solving the linear differential condition (3.35)
L-$=0,
whose solution $0 yields the state of lowest angular momentum in a given energy level. States degenerate (in energy) with this state tiO, but with higher angular momentum, are then obtained by acting on e0 with the raising operators L, (from (3.32) it is clear that L, each commute with the Hamiltonian). The regular solutions to (3.35) with integer angular momentum are -” $o=
(I+
,z,:,2g)“+~’
n = 0, 1, 2, ..*.
(3.36)
To compare with the states g$/, in (3.21) we first extract the measure factor l/( 1 + 1~(‘/2g)~ and note that from (3.21)
-II
Oc (1 + ,:iZ,2g)f19
(3.37)
where in the last step we used the definition (3.20) of the Jacobi polynomials. So, as expected, the states in (3.36) correspond to the lowest angular momentum states (j= -n) in the n th “Landau level.” The n th “Landau level” is then tilled out by
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acting on tiO with L, . In the monopole system each level has a finite degeneracy because (L+YK+2n+*
(1 +
(3.38)
/ ,t/2g)a+,,=o. z
(In the Landau system the degeneracy of each Landau level is infinite because there is no solution to 6’$ = 0.) In the monopole system one could, of course, first determine the state of highest angular momentum in each “Landau level” by solving L+$=O.
(3.39)
The regular solutions, with integer angular momentum,
are
y+b *=
(1 + 1”12/2g)“+“’ 1 Oc (1 +
n=o,
1,2, ...
(R$.pzf%)~
Izl’/2g)R
The degenerate states of lower angular momentum are then obtained by acting on $ with the Lp operator. Having determined that the lowest angular momentum state $,, has angular momentum --n (and hence has L, eigenvalue = -n - g), the energy of this state (and hence of the whole “Landau level”) follows immediately from (3.34) as
E=;(O+(-n-g)(-n-g- 1)--g2) =B n+I+H(n+l) ( 2 2g > ’
(3.41)
which agrees with the spectrum found in (3.21) from the differential equation approach. The relationship between this algebraic formulation of the monopole problem and the algebraic formulation of the Landau problem is one of contraction [23] of Lie algebras. Indeed, defining the resealed generators
(3.42a)
250
GERALD
V. DUNNE
(3.42b)
(3.42~) the commutation
relations (3.3 1) become
Comparing with the Landau generators in (2.27) we see that as g -+ co (recall from (3.7) that this is the infinite radius limit which keeps the magnetic field strength B fixed ), c!Y++b+
(3.44)
Lie -rb
223-+ -II, and the SU(2) commutation commutation relations
relations in (3.43) contract
to the Heisenberg algebra
[b+, b] = --II
(3.45)
[Q, b] = 0 = [I, b+].
The Hamiltonian
operator (3.34) becomes
= B
2’+Yp
-J+;+$-
(
=
1) >
B(b+b-J+;)
(3.46) which is the Landau Hamiltonian
(2.25).
HILBERT
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251
The algebraic discussion thus far has been in terms of operators which act on the wavefunctions I/I which include the measure factor l/( 1 + \z12/2g)g. The corresponding operators which act on the Hilbert space elements $ (see (3.12)) are obtained by conjugation
2+= l+lZ(ZgT 1 + (1 + lr12/2g)R ( 2g) -L
(3.47a)
(3.47b)
p=(l+lr/zgy l %1 3 (1 + Iz(*/2g)g 3
= Y3.
(3.47c)
With respect to the inner product (3.14), p+ and g- are adjoints, while 9’ is selfadjoint. Using the difference and differential-difference relations for the Jacobi polynomials listed in the Appendix, one may check that these operators act on the wavefunctions g$; as J2+(“$‘,)=J(2g+n-j)(j+n+ ~-(g~‘,)=J(2g+n-j+l)(j+n)/2g(g~‘,-’),
1)/2g (q’,+1),
(3.48a) (3.48b) (3.48~)
This confirms the role of the J?~ operators as step operators for the angular momentum index j. Note that since the dependence of the energy on the “Landau” level index n is nonlinear, there is no simple analogue of the energy step operators, ut and a, which appeared in the Landau system. To conclude this discussion of the algebraic structure of the monopole system’s Hilbert space, we show that the wavefunctions g$i, have a simple representation in terms of matrix elements of coherent states for the SU(2) group. Such coherent states are defined as follows [19]. Using the generators L,, L3 satisfying the Chevalley form (3.31) of the SU(2) Lie algebra, we first construct the finite-dimensional spin s representations (of dimension 2s + 1) in the usual way. Define the “vacuum” Js, -s) state annihilated by L _, L- Is, -s)=O,
(3.49)
252
GERALD
V. DUNNE
and excited states. (3.50)
Then the generators act within this representation as
L3 I-J,P> =P 1%II>,
(3.51a)
L+ IkP>=J(wL)(s+p+l)
Is,p+l),
L I~,~>=J(~+11)~~-~+~)l~,~--1).
(3.51b) (3.51c)
The coherent state I[) is then defined as (3.52)
Ii> = Wi) I% -s>, where D(c) is the coherent state operator
= exp “‘“;;y i
‘5’ (CL, -[L-)1.
The matrix elements of the coherent Ref. [24], and give
state operator
x p~,r’-,Pd’+Pl
3-G
to rescale [ by l/A
=exp(zY+)exp exp(zb+) exp
l+* (
_ !$
(
in
l- ICI2 iTp+ ( >
(3.54)
and consider
gln (
q
have been computed
(s+$)! (s-p’)! i”‘-” (s+p)! (s-p)! (l+ 1112)“’
(s, $I D(i) 1%P) =
It is now appropriate
(3.53)
Y 2J
exp( -yb),
>
exp(-zZ) 3)
(3.55)
which should be compared with the coherent state operator (2.35) for the Landau system. From (3.54) we see that
HILBERT
SPACE
FOR
CHARGED
q--Fm(2g+n-j)!
PARTICLES
-’ (n-tj)!
253
p))2g-J,(; ; Lm)
(2g)’ (1+ ,;1’,2g)e (3.56)
It is natural that the wavefunctions in the nth Landau level are given in terms of matrix elements in the spin (n + g) representation as we already saw in (3.35)-(3.38) that the L, , L, generate representations of N(2) of dimension 2n + 2g + 1.
IV.
SYNTHESIS
In this section we bring together the results of the Landau and monopole systems into a unified framework. This then suggests another natural deformation of the Landau system, which we shall call the hyperbolic monopole model. The essence of these models is that we consider charged particles constrained to move on a two-dimensional surface in the presence of a uniform magnetic field which is everywhere “orthogonal” to the surface. Here the word “orthogonal” presumes some knowledge of an embedding of the two-dimensional surface into a three-dimensional space. To express this in intrinsically two-dimensional language we simply say that we consider charged particles in two dimensions coupled to a gauge field whose field strength two-form F=F,,dzdT
is proportional
(4.1)
to the surface area two-form d=S = G,, dz dZ,
(4.2)
where G is the metric on the two-dimensional surface. The plane and the sphere are special in the sense that we can express the metric in terms of a Ktihler potential @, G,, = dd@,
(4.3)
where plane sphere (radius R = ,,&@).
(4.4a) (4.4b)
254
GERALD
V. DUNNE
If we choose a gauge field with components
(4.51 then the corresponding
field strength is
F;, = a&D, = G,:,
(4.6)
(Note that the positive proportionality factor, B, has been absorbed into the complex coordinates, as in (2.3) and (3.9)) The Hamiltonian on an arbitrary surface is given by
where v--iv--A.
(4.7)
Using the Kahler metric in (4.3) and the gauge potentials in (4.5) this reduces to
H=- --&(a-a@)(d+i3@).
(4.8)
The synthesis of the Landau and monopole systems arises from the fact that the respective Hamiltonians (2.4) and (3.10) have the common form (4.8). Indeed, the Landau Hamiltonian (2.4) and the monopole Hamiltonian (3.10) may each be rewritten as
Hz- Bca-a@)(d+d@)+$ 2adG
where the Klhler
potential
@ is given in (4.4a), (4.4b) and the zero-point energy
B/2 arises from a normal ordering which is necessary because of the non-commutativity of the velocity operators in (4.7). Given a Hamiltonian of the form (4.9), it is natural to redefine the wavefunctions $(z, Z) = eC@tj(z, if), which should be compared to (2.6) and (3.12). The “reduced” Hamiltonian
(4.10) becomes
fi=e@He-@ = --f-(a-2aqiJ+f.
2aBG
(4.11)
HILBERT
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PARTICLES
255
From (4.11) we observe the existence of a “lowest Landau level” of energy B/2, described by holomorphic functions-i.e., functions f such that d f= 0. The inner product on the Hilbert space is (fl g) cc J d’seC’@f(z, 5) g(z, 5).
(4.12)
where the overall normalization is chosen so that (111) = 1. With the Kahler potential @ a function of Iz(‘, fi, given as in (4.1 l), commutes with J. Thus, as before, we may seek simultaneous eigenstates of the form $ = z’P( I:( ‘). The energy eigenstate condition equation for P(X),
(4.13)
for I,$ then becomes the ordinary
differential (4.14)
where x reduces Taking equation
= JzI’ and prime ’ means d/dx. Taking @(lzl’) = IzI’, as in (4.4a), this to the Laguerre differential equation (2.13) of the Landau problem. @()~(~)=2gln(l+ 1z12/2g), as in (4.4b), (4.14) reduces to the Jacobi (3.18) of the monopole problem after redefining the argument of P to be
Cl- l~12/%Yu + l#/k). This formulation in terms of Kahler metrics on two-dimensional surfaces suggests another model which is a different deformation of the Landau system. Choosing the Kahler potential @ to be @(lzl*)= the Hamiltonian
-2gln(l-
IzI’/2g),
(4.15)
(4.9) becomes
The subscript “HM” signifies that this is the stereographically projected Hamiltonian for charged particles in the presence of a “hyperbolic monopole.” By “hyperbolic monopole” we shall mean a magnetic field in three-dimensional Minkowski space which is everywhere orthogonal to the upper sheet (x3 9 R) of the hyperbolic surface
-(~~)~-(.x~)~+(x~)~=R~.
(4.17)
Using hyperbolic coordinates (p, z, d), .u’=psinhtcos~ .x’ = p sinh r sin 4 x3 = p cash z,
(4.18)
256
GERALD
V. DUNNE
we may represent a hyperbolic monopole (in the upper sheet with x3 30) (positive) charge g by means of the vector potential
A=_P(P+-Y3)‘p(p+x3)’ gx2 gx’ o. c J The corresponding
of
(4.19)
magnetic field is
B+,
(4.20)
P2
where fi = (sinh r cos 4, sinh z sin 4, -cash r) is the unit normal (in the Minkowski sense) to the hyperbolic surface. When the motion is restricted to the hyperbolic surface p = R (and x3 3 0), the quantum mechanical Hamiltonian is ~,,=1(~x(-id-A))~l~=R.
(4.21)
To understand the relationship between the hyperbolic monopole system and planar Landau system (and also the spherical monopole system) we transform stereographically projected coordinates. This projection maps the upper sheet of hyperboloid to a planar disc of radius &R-see Fig. 2-and is achieved by following change of variables:
the to the the
(4.22)
x3=0
FIG. 2. A cross section, at fixed polar angle (, of the stereographic projection of the point P on the surface of the upper sheet of the hyperboloid of radius of curvature R from the vertex of the lower sheet to the point P’ lying in the tangent plane to the hyperboloid at the vertex of the upper sheet. The hyperbolic surface projects to a planar disc of radius 2R, and the distance JzI is 1.~1= 2R tanh(r/2).
HILBERT
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PARTICLES
257
Once again, the JB/2 factor is introduced to conform with the normalizations in (2.3) and (3.9). Recall that on the hyperbolic surface, p = R, the magnetic field has magnitude (4.23) B = g/R’. Thus, from (4.22) we see that (=I2 < 2BR2 = 2g
(4.24)
and hence we may consider the projected disc to be of radius A. In terms of these coordinates, the hyperbolic monopole Hamiltonian (4.21) becomes ~~~=B(-(~-~)‘?8~(,-~)(~~-id)+tlr’).
(4.25)
As in the spherical monopole case, the limit in which the radius of curvature R -+ 00 in such a way that B remains fixed may be considered simply as the dimensionless limit g + co. Then H,, clearly reduces to the Landau Hamiltonian (2.4) as g --f co. From (4.24), the disc described by the coordinates z, z covers the entire plane in the limit g-co. The Hilbert space of eigenstates is defined as in (4.10) and (4.12). Defining “reduced” wavefunctions I,&, 3) by (4.26) the inner product for the $ is
(4.27) 2g- 1 1-@ 2g-2dqdf, 4nig ( & ) L.
L+(z) = -
Whenf and g are restricted to the holomorphic functions, this is the standard inner product for the Hilbert space of holomorphic functions on the disc-see Bargmann’s treatise [25] on the representation theory of SO(2, 1). The reduced Hamiltonian operator (4.11) is
1 Au”=(1-,Z,2/2g)gHHM (4.28)
258
GERALD
V. DUNNE
Since A,, commutes with the angular momentum wavefunctions of the form
generator J = zd - 58, we seek
tj = z’P( cash t ), where cash t =
1 + Jz12/2g 1 - )z)2/2g’
(4.29)
The energy eigenvalue equation for $ then leads to the following ordinary differential equation for the function P(x): (I-x2)~+2(-g-i-(-g+I)x)~-2g(~-~)P=o.
(4.30)
Note that this is exactly the same differential equation (3.18) as appeared in the spherical monopole problem, with g replaced by -g. There is, however, a crucial difference. Here, the variable x ranges between 1 and co; in (3.18) x ranges between - 1 and + 1. Regular solutions to (4.30) are called associated Jacobi functions [ 161 and lead to wavefunctions (4.31) with energy (4.32)
The wavefunctions (4.31) are orthonormal [16] with respect to the inner product (4.27). There is also a continuous part of the spectrum-for a complete discussion see Refs. 26 and 27. It is instructive to analyze these wavefunctions in an algebraic formalism, as in Section 1I.B for the Landau system and Section 1II.B for the monopole system. For the hyperbolic monopole system the relevant algebra is the Lie algebra of SU(l, 1) (or SO(2, 1)). Indeed, defining the differential operators 1
K,=--
z2a+&a-&L
(4.33a)
& K-=-&d+
1
-zr’d-Jg/2;
(4.33b)
A K3=za-z&gg,
(4.33c)
HILBERT
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FOR
CHARGED
259
PARTICLES
these are generators of SU( 1, 1 ), [K,,
K-1 = -2K3
CK3,K*l==+K+. The hyperbolic monopole Hamiltonian (4.16) may be expressed in terms of the quadratic Casimir operator for SU( 1, 1 ), H
;(K,K+K-K+)-K:+g'
From (4.33) and (4.34) we see that the operators K, and K- act as raising and lowering operators (respectively) for angular momentum. The spectral problem may now be solved by first solving the linear differential condition
(4.36)
K-$=0,
whose solution t+kOis the lowest angular momentum state in a given “Landau level.” States degenerate (in energy) with $0, but with higher angular momentum, they are obtained by acting on $O with the raising operators K,. The regular solutions to (4.36) with integer angular momentum are
~o=~~(l-~)g-‘t, n=o,1,2
,....
(4.37)
In contrast to the (spherical) monopole case, here the “Landau levels’” are infinitelvv degenerate because successive applications of K, to tiO never produces zero (contrast to Eq. (3.38)).* Solving K, 41,= 0 leads to solutions which are either singular or al! have negative energy. Rewriting H HM=;(K+K--K#-l)+g'), we see that I//O (and hence the entire Landau level generated from it) has energy E=$O-(-n+g)(-n+g-
l)+g2)
(4.39) as noted in (4.32). Note that for a given g there exists some n for which the energy becomes negative. ’ Solving K, II/ = 0 leads to solutions which are either singular or all have negative energy.
260
GERALD V.DUNNE
In the limit g + co, the energy spectrum (4.39) reduces to the Landau spectrum and the resealed generators
contract to the Heisenberg generators bt, h, and 21,respectively.
V. CONCLUSIONS We have shown how to construct the complete single-particle Hilbert space for charged particles in an external, perpendicular magnetic field in the following twodimensional configurations: the planar Landau system, the spherical monopole system, and the hyperbolic monopole system. In terms of appropriately chosen stereographically projected complex coordinates the latter two systems reduce simply to the planar Landau model in the limit of infinite monopole charge (this limit corresponds to the infinite radius of curvature limit such that the magnetic field strength at the surface remains fixed). We have also shown that these systems are particular cases of the problem of particles constrained to a two-dimensional Kahler manifold, interacting with a gauge field whose field strength two-form is proportional to the surface’s Kahler form. In this sense we may think of the particles as interacting directly with the surface itself. The three cases treated in detail (the plane, the sphere, and the hyperboloid) are each special for two reasons. First, due to the symmetry of the surfaces about the x3-axis, there is a globally defined angular momentum operator J= ~8 - 58 which commutes with the Hamiltonian-this makes the diagonalization of the Hamiltonian simpler. Second, each of these surfaces has constant curvature (the plane has zero curvature, the sphere has constant positive curvature, and the hyperboloid has constant negative curvature). Correspondingly, there exists, in each case, an algebraic formulation of the spectral problem which effectively reduces it to the problem of solving a linear (rather than quadratic) differential condition. This produces a “Landau level” type structure, with degenerate (in energy) states being created by use of step operators for angular momentum. Note, however, that in the hyperbolic case the relevant symmetry algebra is noncompact and consequently there is also a continuous part in the spectrum (see Refs. 26 and 27 for details). Direct solutions of the energy eigenvalue equation may also be given in terms of Laguerre polynomials, Jacobi polynomials, and associated Jacobi functions for the planar, spherical, and hyperbolic systems, respectively. In the planar and hyperbolic systems the degeneracy of each “Landau level” is infinite, due to the infinite area of the surface; whereas, in the spherical system the degeneracy is finite, due to the finite area of the spherical surface. A common feature of all three models is the existence of a “lowest Landau level” of degenerate states of energy B/2. In each case the lowest Landau level states are holomorphic functions of the corresponding complex coordinate. This means that much of the
261
HIL~ERTSpACEF~RCHAR~ED PARTICLES
analysis of multi-particle states in the lowest Landau level of the planar system may be directly translated to the other systems simply by changing the Hilbert space measure factors. For example, the Laughlin [2] states
have analogues in the spherical geometry which were written down by Haldane in terms of “‘spinor coordinates” [3]. A more direct comparison arises in the stereographically projected coordinates, in which the Haidane states are
Note that in the spherical monopole system the lowest Landau level has finite degeneracy ( = 2g -+ 1 ), and so a wavefunction describing a tilling fraction v = 1/(2m -t 1) presumes a relation between the particle number N, the monopole charge g and v as g = $(N/v - 1).
APPENDIX:
PR~P~T~~~F
The Laguerre polynomials
LAGUERREAND JACOBIPOLYNOMIALS
LIpl’(x) satisfy the di~erential-difference
equation:
(A.1)
x $ L?‘(x) = nLff’(x) - (n + cc)LFL l(X). Tbe L::‘(x)
also satisfy the following difference relations: Lf.y ‘1 = L:‘(x)
- L;:l L(.x),
fA.2)
xL~+“(xt=(x-n)Lf:‘(xf-t(a+n)1;~~,(.~f.
(A.3)
The Jacobi polynomials
PF*p’(x) satisfy the differential-difference
equation:
(A.4)
262
GERALDV.D~NE
Some authors (see, e.g., Ref. [16]) define related Jacobi polynomials
Cm+nVZp)“,“.“+“‘(x).
x(1+x)
Pi,(x)
as
(A.9)
The Pk,(x) satisfy the differential equation
= -Z(Z+ l)f$,
(A.lO)
and have the integral representation
(A.1 1)
m-f-1 du,
where the contour r in (A.1 1) is over the unit circle /WI = 1. The Jacobi functions @k=(x) are also solutions to the differential equation (A.lO), but are defined on the interval XE [l, co), rather than the interval XE [- 1, t] on which the Pf,,{x) are defined. The &,(.x) have the following integral representation Pf?,,(cosh z) =&
$r (cash i + o sinh i)“’ I-n ocoshi+sinhi
urn-‘-
’ da,
(A.12)
>
where the contour r in (A.12) is over the unit circle (~1 = 1.
I am grateful to A. Lerda and A. Perelomov for helpful discussions and comments and to the Aspen Center for Physics where this work was completed. REFERENCES 1. For an excellent
review,
see S. GIRVIN
AND R. PRANGE,
“The Quantum Hall Effect,” Springer-Verlag,
New York, 1990, and references therein. 2. R. LAUGHLIN, Phys. Rev. &if. 50 (1983), 1395; Phys. 3. F. D. M. HALDA~%, Pb.vs. Ret>. Left 51 (19831, 605. 4. P. DIRAC, Proc. R. Sot. London A 133 (t931), 60.
Rev. B 29 (1983),
3383.
HILBERT
SPACE
FOR CHARGED
PARTICLES
263
5. I. TAMM, Z. Whys. 71 (1931). 141. 6. M. FIERZ, Helv. Phw. Acta 17 (1944), 27. 1. T. T. Wu AND C. N. YANG, Nucl. Phys. B 107 (1976), 365. 8. F. D. M. HALDANE AND E. M. REZAYI, Phys. Rev. Letf. 54 (1985), 237. 9. L. LANDAU, Z. Phys. 64 (1930), 629; L. LANDAU AND E. LIFSCHITZ,
10. 11. 12. 13. 14.
15. 16. 17.
“Quantum Mechanics: Non-Relativistic Theory,” Pergamon, New York, 1977. S. GIRVIN AND T. JACH, Phys. Rev. B 29 (1984). 5617. G. FANO, F. ORTOLANI, AND E. COLOMBO, Phys. Rev. B 34 (1986), 2670. V. BARGMANN, Rev. Mod. Phys. 34 (1962), 829. R. LAUGHLIN, SurJ: Sci. 141 (1984). 11. M. ABRAMOWITZ AND I. STEGUN,“Handbook of Mathematical Functions,” Verlag Harri Deutsch, Thun, Switzerland, 1984. G. SZEGB, “Orthogonal Polynomials,” Amer. Math. Sot., Providence, RI, 1967. N. VILENKIN, “Special Functions and the Theory of Group Representations,” Amer. Math. Sot., Providence, RI, 1968. Such parametric limits in magnetic systems have been studied previously in G. DUNNE, R. JACKIW, AND C. TRUGENBERGER,
C. 19. A.
Phys.
Rev. D 41 (1990),
661.
M. JOHNSON, Phys. Rev. B 41 (1990), 6870; G. DUNNE, A. LERDA, S. SCIUTO, AND TRUGENBERGER, MIT Preprint CTP 1978, June 1991, to appear in Nuc~. Phy.7. B. F’ERELOMOV. “Generalized Coherent States and their Applications,” Springer-Verlag. New York.
18. G. CANRIGHT
1986. 20. J. SCHWINGER.
AND
Phys.
Rev. 91 (1953).
728.
21. S. Iso, University of Tokyio Preprint UT-579, February 1991. 22. A. ERD~LYI (Ed.), “Higher Transcendental Functions, Vol. II, Bateman Manuscript Project,” McGraw-Hill, New York, 1953. 23. R. GILMORE, “Lie Groups, Lie Algebras, and Some of Their Applications,” Wiley, New York, 1974. 24. A. EDMONDS. “Angular Momentum in Quantum Mechanics,” Princeton Univ. Press, Princeton, NJ, 1974. 25. V. BARGMANN, Ann. Math. 48 (1947) 568. 26. A. COMTET, Ann. Phys. (N.Y.) 173 (1987). 185. 27. C. GROSCHE,
595121512-3
Ann.
Phys.
(N.Y.j
187 (1988).
110.
OF PHYSICS
215, 233-263 (1992)
Hilbert Space for Charged Particles in Perpendicular Magnetic Fields* GERALD Department Massachusetts
V. DUNNE
of Mathematics and Center ji)r Theoretical Physics, Institute of Technology, Cambridge, Massachusetts 02139 Received October 9. 1991
We describe the quantum mechanics of two-dimensional charged particles in a perpendicular magnetic field in the planar Landau and spherical monopole configurations. These models, particularly in the work of Laughlin and Haldane, are crucial to the theoretical understanding of the quantum Hall effect. Here we present the full Hilbert space structure in each case, with special emphasis on the relationship between the two systems. The formulation in terms of stereographically projected complex coordinates makes the connection especially explicit and naturally generalizes to more complicated two-dimensional surfaces where the interaction of the particles with an external perpendicular magnetic field may be regarded as an interaction of the particles with the two-dimensional (Kahler) metric of the surface. This generalization is illustrated by the hyperbolic configuration of particles constrained to the upper sheet of a hyperboloid in the presence of a hyperbolic monopole. ‘c 1992 Academic Press. Inc.
I. INTRODUCTION
Great advances (both experimental and theoretical) have been made in recent years in the understanding of two-dimensional systems of non-relativistic charged particles. One such advance, the quantum Hall effect, involves a two-dimensional system of electrons in a perpendicular external magnetic field. The quantum behavior of such a system, especially at very low temperature and very high magnetic field strength, holds many interesting surprises [l]. A major theoretical step was taken with Laughlin’s proposal [2] of variational wavefunctions describing incompressible states corresponding to special rational filling fractions of the highly degenerate lowest Landau level. Haldane [3] introduced a variant of the Laughlin wavefunctions by considering a system of electrons constrained to the surface of a sphere with a magnetic monopole at the center of the sphere. The singleparticle states for such a system were originally studied by Dirac [4], Tamm [S], and Fierz [6] and are essentially the “monopole harmonics” of Wu and Yang [7]. A significant advantage of the spherical geometry system is that it permits transla* This work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under Contract DE-AC02-76ER03069. and NSF Grant 87-08447.
233 0003-4916/92 $9.00 CopyrIght ,(‘I 1992 by Academic Press. Inc. All rights of reproducfmn m any form reserved.
234
GERALD
V. DUNNE
tionally and rotationally invariant states with finite particle number and is especially suited to numerical simulations [3, 81. The finite area of the spherical surface translates into finite degeneracies of the energy levels, and this allows for a simpler treatment of the thermodynamic limit. In this paper we describe the full Hilbert space structure of the planar Landau system [9] and of the spherical monopole system, and we describe precisely the relationship between the two models. The monopole system is a consistent deformation of the planar system, with the deformation parameter being the inverse of the monopole charge g. In the limit g + cc all features of the Landau model are regained from the corresponding features of the monopole model. The magnetic field strength at the spherical surface (of radius R) due to a monopole of charge g is B = g/R2, so the limit g + co corresponds to the infinite radius limit in which B is kept fixed. Previous discussions of the Landau [lo] and monopole [3,8, 1 l] systems in the context of the quantum Hall effect have concentrated on the lowest Landau level-here we incorporate all excited states into the Hilbert space. This is motivated by the need to study interactions between the electrons (especially at higher temperatures and in weaker B fields) and also by the obvious desire for theoretical completeness. We show that by a suitable choice of stereographically projected coordinates the relationship between the Landau and monopole systems becomes explicit and transparent. The full Hilbert space in each case may be found either by solving a second-order linear ordinary differential equation or by making use of related algebraic structures to reduce the problem to a first-order linear differential condition. In the planar case the algebraic structure arises from the Heisenberg algebra of non-commuting magnetic translations [9], while the relevant algebraic structure in the spherical case in SU(2) [S-7]. The wavefunctions, in each case, can be expressed as matrix elements of the coherent state operator for the corresponding algebra. We also show that these two models are special cases of the more general model of charged particles constrained to the surface of a two-dimensional Kahler manifold, interacting with an external gauge field which has its field strength two-form proportional to the surface’s Klhler form (this says that the particles move in a uniform magnetic field per unit area which is everywhere normal to the surface). Another example of this structure is that of particles constrained to a hyperboloid in the presence of a hyperbolic monopole. This paper is organized as follows. In Section II.A, the full Hilbert space is described for the Landau system of planar charged particles in the presence of a constant normal magnetic field. In Section 1I.B we discuss the algebraic step operator approach to this and show that the wavefunctions are matrix elements of the coherent state operator for the Heisenberg algebra. The Hilbert space structure of the spherical monopole system is derived in Section IIIA, and the corresponding algebraic analysis is given in Section 1II.B. Here the wavefunctions are matrix elements of the SU(2) coherent state operator. In Section IV we discuss a synthesis of these models in terms of two-dimensional Kahler manifolds and illustrate this synthesis with another related model, that of the hyperbolic monopole. Here the relevant algebraic structure is the (non-compact) Lie algebra of SU(1, 1). In this
HILBERTSPACEFOR
CHARGED PARTICLES
235
paper we restrict our attention to the discrete part of the spectrum in the hyperbolic system. For a complete discussion of this model, including the continuous part of the spectrum, see Refs. 26 and 27. An Appendix contains some properties of the Laguerre polynomials, Jacobi polynomials and associated Jacobi functions.
II. LANDAU
SYSTEM
A. Hilbert Space In this section we review the Landau problem [9] of non-interacting charged particles moving in the two-dimensional plane in the presence of a fixed external magnetic field (of strength B> 0) perpendicular to the plane. The quantum mechanical Hamiltonian operator for this system is (we work in units ti=m=e=l)
=$(-i8-A)2,
(2.1)
where we choose the vector potential A in the symmetric gauge
It proves more convenient to work in terms of the resealed complex coordinates z=$iji(xl
+ix2)
‘1 In these coordinates the Landau Hamiltonian
(2.1) is
H,=B(-~~-&-~~)+~z~),
where 8 = a/&, 8 = a/&. The planar angular momentum J=zd-Z(q
(2.4) operator (2.5)
commutes with H,. We shall therefore seek simultaneous eigenstates II/ of energy and angular momentum. To define the single-particle Hilbert space, we extract an exponential measure factor $(z, 5) = e-(1P) l=l*$(z, 2).
(2.6)
The “reduced” wavefunctions $ will be identified with elements of the Hilbert space,
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while the exponential factor contributes to the measure in the Hilbert space inner product: (2.7a)
In (2.7), f and g are functions of both z and Z. The bar ~ over f in (2.7) denotes complex conjugation, as in (2.3). The inner product (2.7) is a generalization of that considered by Bargmann[ 121 in his description of the Hilbert space of analytic functions on the plane, and which has been applied to the lowest Landau level of the Landau system by Girvin and Jach [lo]. Note, however, that in order to incorporate the higher Landau levels we cannot restrict the functionsfand g to be holomorphic-the appropriate generalization is described below. Having identified the “reduced” wavefunctions I,& in (2.6) with elements of the Hilbert space, one should identify the operators acting in the Hilbert space as those acting on 6, rather than on $.’ The relation between an operator 0 acting on II/ and the corresponding Hilbert space operator 6 acting on I,& is simply one of conjugation d=e
(l/2)
Irl’o,-(l/2)
/=I?.
(2.8)
Thus, the Hilbert space energy and angular momentum
operators are
A,=B(-a8+2++),
(2.9)
.Lza-zLi
(2.10)
The lowest Landau level, of energy B/2, is described by holomorphic wavefunctions lj = l)(z). Now write an arbitrary wavefunction $ as an angular momentum eigenstate $=z-‘P(lz12),
(2.11)
where P is some function to be determined and j is an integer to ensure singlevaluedness of 6. Requiring $ to also be an energy eigenstate, @,$=E$
(2.12)
then leads to the following ordinary differential equation for the function P(x): x$+(j+l-x)$+ ’ As noted by Girvin and Jach [lo] in the context [ 131 procedure of not having the differential operator of the form $ in (2.6).
(
5-t
>
P=O.
(2.13)
of the lowest Landau level, this explains Laughlin’s 8 act on the exponential part of the wavefunctions
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This is a differential equation of Laguerre form, xh” + (j + 1 -x)/r’
+ nh = 0
(2.14)
which has as regular solution the generalized Laguerre polynomial h = L’,(x)
(2.15) Further properties of the Laguerre polynomials are listed in the Appendix. Comparing (2.13) and (2.14) we obtain the solution
$~=&&z-j$z’L’,(Iz12), E=B(n+$),
(2.16)
n = 0, 1) 2, ...) j= --n, --n + 1, ... . The normalization
factor arises from the Laguerre integral [14] ccd.xe-“x’L.;(x)
(n+j)! L’,(x) = 6,, ,r.
(2.17)
.r 0
Note that Lx is defined (and normalizable in the sense of (2.17) being finite) for j>, -n, and when the upper index is negative one may also write (2.18) From (2.16) we recognize n as the familiar “Landau level” index [9] and we see that the Hilbert space 2 decomposes into an infinite number of such Landau levels
each Xn being infinitely degenerate-the degenerate states being distinguished by their angular momentum index j. Note that the lowest Landau level states (i.e., those with it = 0) all have non-negative angular momentum, while the higher Landau levels, Xn with n 2 1, contain states with negative angular momentum. It follows from (2.17) that the wavefunctions 1+6i,in (2.16) are orthonormal with respect to the Hilbert space inner product (2.7). In fact, they also form a complete
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orthonormal set of wavefunctions [15, 161. We can therefore express any element of the single-particle Hilbert space as a linear superposition (2.20)
where the numerical coefficients a,, j must be such that $ is normalizable with respect to (2.7). In certain physical applications (for example, when B is very large [ 171) it is instructive to consider just the lowest Landau level Z. portion of the Hilbert space. As mentioned previously, the above discussion then reduces to Bargmann’s Hilbert space of analytic functions with basis functions I)~=~=-$z~,
j=O, 1,2, ....
(2.21)
which form a complete orthonormal set for analytic functions [12]. (Note that the wavefunctions $i, in (2.16) form a complete orthonormal set for functions of z and 5.) Within & a special role is played by Bargmann’s “principal vectors” exp(wz)
(UEC).
(2.22)
These functions play a role similar to that of the Dirac b-function in the conventional position space representation of quantum mechanics. They arise as a resolution of the identity within the restricted Hilbert space Zo, f
tj’,=,(o)
g’,=,(z) = exp(&),
(2.23)
j=O
and have the important
property
f(u) =f(z), 5b(w) exp(6z)
(2.24)
which illustrates their similarity with the Dirac a-function. B. Algebraic Approach The Hilbert space structure described thus far has a simple algebraic formulation which is of great practical use in explicit computations. (It also provides the key to a systematic analysis of the many-body quantum mechanics of anyons in a magnetic field [18].) The idea is very simple-the energy and angular momentum operators may be expressed in terms of two (commuting) sets of harmonic oscillator step operators, H, = B(ata + i)
(2.25)
J= btb - ata,
(2.26)
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where
(2.27)
Then the a’s and b’s commute with one another, but amongst themselves they satisfy the Heisenberg algebra commutation relations
[a,a+]= n,
[b,b’]=Q.
(2.28)
From (2.25)-(2.28) we see that bt and b have a simple interpretation as raising and lowering operators (respectively) for angular momentum. As they commute with H,, they act within a given Landau level connecting states of equal energy. Similarly, the ut and a act as raising and lowering operators (respectively) for energy, while simultaneously acting as lowering and raising operators (respectively) for angular momentum. They therefore connect states in different Landau levels. The operators atbt and ab act as raising and lowering operators (respectively) for energy, while leaving the angular momentum unchanged. To consider their action on the wavefunctions $ we should examine the conjugated operators Lit, 4, @, 6 defined as in (2.8), $=a-=
(j= -a p=
-$+z
(2.29)
6= a. These operators, of course, satisfy the same commutation relations as the corresponding a’s and b’s, and the relations (2.25) and (2.26) simply acquire “hats” (“““) throughout. Moreover, with respect to the inner product (2.7) it is easy to verify that ri and tit (and also 6 and !?) are truly adjoints, i.e., (2.30)
The action of these Hilbert space operators on the wavefunctions $j, in (2.16) is (2.31a) (2.31b) (2.31~) (2.31d)
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GERALD V.DUNNE
In verifying (2.31a)-(2.31d) we use the differential-difference and difference relations for the Laguerre polynomials listed in the Appendix. Note also that the “bottom ” of the nth Landau level is the angular momentum state with j= --n because &$L= M-n= 0. The n th Landau level is unbounded above, since $i, is never annihilated by 6’. We conclude this discussion of the algebraic structure of the Landau system’s Hilbert space by showing that the wavefunctions 4; have a simple representation in terms of matrix elements of coherent states for the Heisenberg-Weyl group. Define a “vacuum” state IO) such that b(O)=0
(2.32)
and excited states by (2.33)
One can then define the coherent state 1~) by [19]
lz> = Wz) IO>,
(2.34)
where D(z) is the coherent state operator D(z) = exp(zb+ - Fb) :b+ -Z/I =e -(I/Z) (iI ee.
It is then straightforward (n+jl
(2.35)
to compute that [20] D(z) In) =~~e-1’2”‘2~iLi,(lzJ2) =e
-(l/2)
[=I2 “j $ nr
(2.36)
which are the Landau wavefunctions, including the exponential measure factor. Various authors [ 10, 211 have noted this correspondence between coherent states and the Landau wavefunction in the lowest Landau level (n = 0), where they argue that the two-dimensional configuration space behaves as a phase space due to the very strong B field which energetically isolates the lowest Landau level. In fact, the correspondence to coherent states generalizes very simply to higher Landau levels, as shown by (2.36). III. MONOPOLE SYSTEM A. Hiibert Space In this section we analyze the quantum mechanics of charged particles constrained to the surface of a sphere with a magnetic monopole at the center of the
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sphere. This problem was originally considered by Dirac [4], Tamm [S], and Fierz [6] and has recently been revived in the context of the fractional quantum Hall effect by Haldane [3]. Here, we reformulate the system in a form which permits an immediate and simple comparison with the Landau model discussed in Section II. The qualitatively analogous features of the two systems are clear--each refers to quantum mechanical charged particles moving on a two-dimensional surface in the presence of a uniform, everywhere perpendicular magnetic field. Intuitively, one would expect that as the sphere’s radius becomes very large the spherical surface becomes “essentially flat” so that the monopole system reduces to the Landau model. The purpose of this section is to make this intuitive expectation precise. The monopole system is formulated as follows. Consider the two-dimensional spherical surface to be embedded in three-dimensional Euclidean space, for which we use both Cartesian (x’, x2, x3) and spherical (Y, 8, 4) coordinates x1 = r sin 19cos 4 x2 = r sin 6 sin Q
(3.1)
x3 = r cos 8.
We represent a magnetic monopole of (positive) charge g, centered at the origin in three-space, by a vector potential expressed in the “Dirac string” gauge,
A=( -$p,&q,~).
(3.2)
The components of the vector potential along the polar unit vectors i, 4, and 4 are A,=A,=O (3.3) A,= --&(1-c0”0).
Note that this vector potential has a singularity along the negative x3-axis-the significance of this will be discussed later in relation with the planar Landau model. The magnetic field corresponding to the vector potential in (3.2) (or (3.3)) is directed radially outwards from the origin: B=VxA
The Hamiltonian governing the motion of charged particles in three-space in the presence of the monopole is
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(3.5)
Restricting the motion to the surface of the sphere r = R, we obtain the required two-dimensional monopole Hamiltonian fz,++ia-A))‘,,=, 1
-- i
a
2R2 sin 6 80 sin
a e-86
1
a?
2R2 sin’ 0 @ (3.6)
At the surface of the sphere the magnetic field is everywhere perpendicular surface, and of constant magnitude: B = g/R’.
The Dirac quantization g to satisfy
condition
to the
(3.7)
[4] requires the dimensionless monopole charge 2g = integer.
(3.8)
The key to understanding the relationship between the monopole system and the Landau system is to transform to stereographically projected coordinates which project from the “South pole” (0, 0, -R) of the sphere to the tangent plane to the sphere at the “North pole” (0, 0, Rksee Fig. 1. In such a projection, the South pole itself is mapped to the point at infinity in the projected plane, so it is in fact more appropriate to talk of the monopole system with Hamiltonian (3.6) defined on the spherical surface r = R with the point at the South pole excluded. As we shall see, one coordinate patch, with the South pole excluded, is sufficient for the purpose of comparing with the Landau system. This explains why it is sufficient to use the vector potential A in the Dirac string gauge (3.2)-to consider the model on the entire spherical surface would require (at least) two patches, a system for which the Wu-Yang [ 71 formulation is more appropriate. The aforementioned stereographic projection is achieved by the coordinate transformation
(3.9)
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N-lz+---
P’
S
FIG. 1. A cross section, at fixed polar angle 4, of the stereographic projection of the point P on the surface of the sphere of radius R, from the South pole S to the point P’ lying in the tangent plane to the sphere at the North pole N. N is mapped to the origin of the tangent plane and S is mapped to the point at infinity. The distance (11 is (z( = 2R tan(0/2).
The extra resealing factor JB/2 has been introduced to conform with the normalization of the complex coordinates (2.3) used in Section II for the Landau system. In terms of these coordinates, the monopole Hamiltonian (3.6) becomes
H,=B _ *+Iz(z2a&’ I+!?!! (z&q++2 (LJ
2LJ
).
(3.10)
Now consider the limit in which the spherical radius R + cc. From (3.7) it is clear that in order to maintain a constant magnetic field B one must simultaneously perform the limit g -+ co (in such a way that B = g/R2 remains fixed). Therefore, the scale R may be eliminated in favor of B, the constant dimensionful quantity, in which case the desired limit is simply that of letting the dimensionless charge g -+ co. In this limit, it now follows from (3.10) that H,-tB(-a~-1(za-i~)+flz12)
(3.11)
which is precisely the Landau Hamiltonian (2.4). To determine the Hilbert space of eigenstates for the monopole Hamiltonian (3.10) we proceed exactly as for the Landau model in Section II. First extract a measure factor from the wavefunctions (at this point, this may be viewed simply as a convenient transformation to simplify the eigenvalue problem) by defining 1 bhz, -3 = (1 + JzI2/Zg)” ij(z, 2).
(3.12 )
1 - lzl2/2 ,‘EZ (1+ ,Z,2,2g)g=e ’
(3.13
Note that
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which is the same measure factor as was extracted from the Landau wavefunctions-see (2.6). Once again, the “reduced” wavefunctions 4 will be identified with elements of the Hilbert space, and the other factor in (3.12) contributes to the measure in the Hilbert space inner product: (3.14a) dz d.5 2g+l d/i(z) = ___ 4ing (1 + 1z/2/2g)28+2’
Then the reduced Hamiltonian wavefunctions $ is
(3.14b)
operator which acts directly on the Hilbert
space
(3.15)
whose g + cc limit agrees with I?, in the Landau model-see (2.9). The operator j= zd -ia commutes with I?, and generates rotations about the origin in the projected plane (which correspond to rotations about the x3-axis in terms of the original sphere). Since j and I?, commute, we seek simultaneous eigenstates-we therefore decompose $ as in (2.11). However, it is now more convenient to express the undetermined function P as a function of (3.16) rather than simply as a function of (z(*. Thus we write (3.17) The energy eigenstate condition fi, I,$= E$ leads to the following ordinary differential equation for the function P(x): (3.18)
(‘--~‘)~+2(g-~-(g+l)x)~+2g(~-~)p~~. This is a differential equation of the Jacobi form (1 -x2)h0+(p-cr-(M+p+2)x)h’+n(n+a+/?+
l)h=O
(3.19)
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whose regular solution in the interval [ - 1, l] is the Jacobi polynomial h = P;,p’(x)
-=%(l
-x-3
(I-J)‘l$
=~~,(“;“)(~‘3
{(l -X)X+‘r (1 +.#+n) (x- l)‘lP,
(3.20)
(s+ 1)“.
Further properties of the Jacobi polynomials are listed in the Appendix. Comparing (3.18) and (3.19) we obtain the wavefunctions n! (2g+2n+
g$:, =
l)I+z+2g+ 1) 1) T(n+2g-j+
(2g+ 1)(2g)‘T(n+j+
Z,p;%-jj
(i + ~~~~~~),
1)
E=B n+l+n(n+l) ( 2 37 1 n=o,
(3.21)
1,2, ...
j= -n, -n+ The normalization
1, ...) n+2g. factor arises from the Jacobi integral [22]
1 I -I
dx( 1 - X)%(1 + x)b Pa-P) n (x) Fyyx) = s,,
2”f”T(n+cr+l)z+z+fl+l) n! (2n+a+B+ l)z(n+a+~+
1)’
(3.22)
Note also that the Jacobi polynomial Pjla,p)is defined for a >/ -n, /I b - n: this leads to the upper and lower bounds on the allowed values ofj in (3.21). From (3.22) we see that the wavefunctions @jn in (3.21) form an orthonormal set with respect to the inner product (3.14). In fact, this set is also complete [16]. Let us first comment on the energy and angular momentum spectrum in (3.21). There is still a “Landau level” structure in the sense that the non-negative integer n determines the energy. However, the dependence of the energy on n is no longer linear-it has a quadratic part also. But as g + co the quadratic part vanishes leaving an energy spectrum of Landau form. Each “Landau level” (i.e., the set of states all having the same index n) is once again degenerate, with degenerate states being distinguished by their angular momentum index j. However, the degeneracy of each level is now finite, there being an upper bound n + 2g for the angular momentum in the nth “Landau level”: degeneracy(nth level) = 2g + 2n + 1.
(3.23 )
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In the limit g + co the degeneracy becomes infinite and the degeneracy per unit area degeneracy(nth level) 2g + 2n + 1 = area 47cR2
(3.24) which is the familiar Landau result [9]. To understand the behavior of the wavefunctions 5,&j, themselves in the limit g + co, we use the limiting relation between Jacobi and Laguerre polynomials c141, lim Pjl’,@) 1-g = L:(X). 8-m ( P1
(3.25)
Therefore, (3.26) and we see that the monopole wavefunctions (3.21), indeed, reduce to the Landau wavefunctions (2.16) in the limit g+ co. As mentioned in Section II, it is sometimes possible to restrict one’s attention to just the lowest “Landau level,” n = 0. Then there are 2g + 1 states spanning this level
g$;=o= (2g)!/(2g)‘j! The corresponding
(2g-j)!
zi 2
j = 0, .... 2g.
resolution of the identity in the lowest “Landau
=I,;.
(
- 2g >
(3.27) level” is
(3.28)
In the limit g -+ cc, this reduces to exp(zti), agreeing once again with the corresponding Landau result (2.23). Furthermore, using the orthonormality relations
HILBERT
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(3.22), it follows that this resolution of the identity within the restricted (n = 0) Hilbert space, Sdr(w)(l
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acts like a Dirac d-function
(3.29)
+~)2gf(o)=tli),
for any holomorphic function f(z). To conclude this section we note the wavefunctions in (3.21) are precisely the Wu-Yang monopole harmonics [7] re-expressed in terms of the projected coordinates for the coordinate patch covering the spherical surface minus the South pole. Excluding the South pole in the monopole system is natural for comparison with the Landau model, as the South pole corresponds to the planar point at infinity. B. Algebraic Approach Just as in the Landau system, the monopole system has an algebraic formulation which considerably simplifies the spectral problem, The key fact [S-7] is that the Hamiltonian (3.10) may be written (up to an additive constant) as the quadratic Casimir operator for the Lie algebra N(2). One then finds that the nth “Landau level” (of degeneracy 2g + 2n + 1) corresponds to a finite-dimensional spin (g + n) representation of SU(2). All aspects of the Hilbert space may be recast in terms of the representation theory of SU(2). This becomes especially clear in the stereographically projected coordinates. The relation to the Landau problem may be understood as a contraction [23] of the Lie algebra of SU(2) to the Heisenberg algebra. Define the differential operators, (3.30a) (3.30b) (3.3Oc)
These are generators of the Lie algebra of SU(2), in Chevalley form, CL,, L-1 =2L,
C-LL*l= Then the monopole Hamiltonian
(3.31)
s,.
(3.10) may be written as ;(L+Ld+LpL+)+L;-g2
> (3.32)
595!215,2-2
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where C, is the quadratic Casimir operator for SU(2). Also, the angular momentum operator, J, is related to L, by J=L,+g.
(3.33)
From (3.32)-(3.33) it follows that the problem of finding simultaneous eigenstates of J and H, is essentially the same as simultaneously diagonalizing L, and the quadratic Casimir operator C2 for X42). The operators L, act as raising and lowering operators for the eigenvalue of I,-hence they act as raising and lowering operators for angular momentum. They are thus the analogues of the raising and lowering operators ht and b (2.27) in the Landau system. Having expressed the Hamiltonian as in (3.32) it is now trivial to generate the entire spectrum of states. H, may be rewritten H,=;(L+Lp
+L,(L,-
l)-g2).
Expressing a wavefunction + as an eigenstate of angular momentum J (and hence also of L3) we see that we simply have to diagonalize L, Lp . This is achieved by first solving the linear differential condition (3.35)
L-$=0,
whose solution $0 yields the state of lowest angular momentum in a given energy level. States degenerate (in energy) with this state tiO, but with higher angular momentum, are then obtained by acting on e0 with the raising operators L, (from (3.32) it is clear that L, each commute with the Hamiltonian). The regular solutions to (3.35) with integer angular momentum are -” $o=
(I+
,z,:,2g)“+~’
n = 0, 1, 2, ..*.
(3.36)
To compare with the states g$/, in (3.21) we first extract the measure factor l/( 1 + 1~(‘/2g)~ and note that from (3.21)
-II
Oc (1 + ,:iZ,2g)f19
(3.37)
where in the last step we used the definition (3.20) of the Jacobi polynomials. So, as expected, the states in (3.36) correspond to the lowest angular momentum states (j= -n) in the n th “Landau level.” The n th “Landau level” is then tilled out by
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acting on tiO with L, . In the monopole system each level has a finite degeneracy because (L+YK+2n+*
(1 +
(3.38)
/ ,t/2g)a+,,=o. z
(In the Landau system the degeneracy of each Landau level is infinite because there is no solution to 6’$ = 0.) In the monopole system one could, of course, first determine the state of highest angular momentum in each “Landau level” by solving L+$=O.
(3.39)
The regular solutions, with integer angular momentum,
are
y+b *=
(1 + 1”12/2g)“+“’ 1 Oc (1 +
n=o,
1,2, ...
(R$.pzf%)~
Izl’/2g)R
The degenerate states of lower angular momentum are then obtained by acting on $ with the Lp operator. Having determined that the lowest angular momentum state $,, has angular momentum --n (and hence has L, eigenvalue = -n - g), the energy of this state (and hence of the whole “Landau level”) follows immediately from (3.34) as
E=;(O+(-n-g)(-n-g- 1)--g2) =B n+I+H(n+l) ( 2 2g > ’
(3.41)
which agrees with the spectrum found in (3.21) from the differential equation approach. The relationship between this algebraic formulation of the monopole problem and the algebraic formulation of the Landau problem is one of contraction [23] of Lie algebras. Indeed, defining the resealed generators
(3.42a)
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GERALD
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(3.42b)
(3.42~) the commutation
relations (3.3 1) become
Comparing with the Landau generators in (2.27) we see that as g -+ co (recall from (3.7) that this is the infinite radius limit which keeps the magnetic field strength B fixed ), c!Y++b+
(3.44)
Lie -rb
223-+ -II, and the SU(2) commutation commutation relations
relations in (3.43) contract
to the Heisenberg algebra
[b+, b] = --II
(3.45)
[Q, b] = 0 = [I, b+].
The Hamiltonian
operator (3.34) becomes
= B
2’+Yp
-J+;+$-
(
=
1) >
B(b+b-J+;)
(3.46) which is the Landau Hamiltonian
(2.25).
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The algebraic discussion thus far has been in terms of operators which act on the wavefunctions I/I which include the measure factor l/( 1 + \z12/2g)g. The corresponding operators which act on the Hilbert space elements $ (see (3.12)) are obtained by conjugation
2+= l+lZ(ZgT 1 + (1 + lr12/2g)R ( 2g) -L
(3.47a)
(3.47b)
p=(l+lr/zgy l %1 3 (1 + Iz(*/2g)g 3
= Y3.
(3.47c)
With respect to the inner product (3.14), p+ and g- are adjoints, while 9’ is selfadjoint. Using the difference and differential-difference relations for the Jacobi polynomials listed in the Appendix, one may check that these operators act on the wavefunctions g$; as J2+(“$‘,)=J(2g+n-j)(j+n+ ~-(g~‘,)=J(2g+n-j+l)(j+n)/2g(g~‘,-’),
1)/2g (q’,+1),
(3.48a) (3.48b) (3.48~)
This confirms the role of the J?~ operators as step operators for the angular momentum index j. Note that since the dependence of the energy on the “Landau” level index n is nonlinear, there is no simple analogue of the energy step operators, ut and a, which appeared in the Landau system. To conclude this discussion of the algebraic structure of the monopole system’s Hilbert space, we show that the wavefunctions g$i, have a simple representation in terms of matrix elements of coherent states for the SU(2) group. Such coherent states are defined as follows [19]. Using the generators L,, L3 satisfying the Chevalley form (3.31) of the SU(2) Lie algebra, we first construct the finite-dimensional spin s representations (of dimension 2s + 1) in the usual way. Define the “vacuum” Js, -s) state annihilated by L _, L- Is, -s)=O,
(3.49)
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GERALD
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and excited states. (3.50)
Then the generators act within this representation as
L3 I-J,P> =P 1%II>,
(3.51a)
L+ IkP>=J(wL)(s+p+l)
Is,p+l),
L I~,~>=J(~+11)~~-~+~)l~,~--1).
(3.51b) (3.51c)
The coherent state I[) is then defined as (3.52)
Ii> = Wi) I% -s>, where D(c) is the coherent state operator
= exp “‘“;;y i
‘5’ (CL, -[L-)1.
The matrix elements of the coherent Ref. [24], and give
state operator
x p~,r’-,Pd’+Pl
3-G
to rescale [ by l/A
=exp(zY+)exp exp(zb+) exp
l+* (
_ !$
(
in
l- ICI2 iTp+ ( >
(3.54)
and consider
gln (
q
have been computed
(s+$)! (s-p’)! i”‘-” (s+p)! (s-p)! (l+ 1112)“’
(s, $I D(i) 1%P) =
It is now appropriate
(3.53)
Y 2J
exp( -yb),
>
exp(-zZ) 3)
(3.55)
which should be compared with the coherent state operator (2.35) for the Landau system. From (3.54) we see that
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-’ (n-tj)!
253
p))2g-J,(; ; Lm)
(2g)’ (1+ ,;1’,2g)e (3.56)
It is natural that the wavefunctions in the nth Landau level are given in terms of matrix elements in the spin (n + g) representation as we already saw in (3.35)-(3.38) that the L, , L, generate representations of N(2) of dimension 2n + 2g + 1.
IV.
SYNTHESIS
In this section we bring together the results of the Landau and monopole systems into a unified framework. This then suggests another natural deformation of the Landau system, which we shall call the hyperbolic monopole model. The essence of these models is that we consider charged particles constrained to move on a two-dimensional surface in the presence of a uniform magnetic field which is everywhere “orthogonal” to the surface. Here the word “orthogonal” presumes some knowledge of an embedding of the two-dimensional surface into a three-dimensional space. To express this in intrinsically two-dimensional language we simply say that we consider charged particles in two dimensions coupled to a gauge field whose field strength two-form F=F,,dzdT
is proportional
(4.1)
to the surface area two-form d=S = G,, dz dZ,
(4.2)
where G is the metric on the two-dimensional surface. The plane and the sphere are special in the sense that we can express the metric in terms of a Ktihler potential @, G,, = dd@,
(4.3)
where plane sphere (radius R = ,,&@).
(4.4a) (4.4b)
254
GERALD
V. DUNNE
If we choose a gauge field with components
(4.51 then the corresponding
field strength is
F;, = a&D, = G,:,
(4.6)
(Note that the positive proportionality factor, B, has been absorbed into the complex coordinates, as in (2.3) and (3.9)) The Hamiltonian on an arbitrary surface is given by
where v--iv--A.
(4.7)
Using the Kahler metric in (4.3) and the gauge potentials in (4.5) this reduces to
H=- --&(a-a@)(d+i3@).
(4.8)
The synthesis of the Landau and monopole systems arises from the fact that the respective Hamiltonians (2.4) and (3.10) have the common form (4.8). Indeed, the Landau Hamiltonian (2.4) and the monopole Hamiltonian (3.10) may each be rewritten as
Hz- Bca-a@)(d+d@)+$ 2adG
where the Klhler
potential
@ is given in (4.4a), (4.4b) and the zero-point energy
B/2 arises from a normal ordering which is necessary because of the non-commutativity of the velocity operators in (4.7). Given a Hamiltonian of the form (4.9), it is natural to redefine the wavefunctions $(z, Z) = eC@tj(z, if), which should be compared to (2.6) and (3.12). The “reduced” Hamiltonian
(4.10) becomes
fi=e@He-@ = --f-(a-2aqiJ+f.
2aBG
(4.11)
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255
From (4.11) we observe the existence of a “lowest Landau level” of energy B/2, described by holomorphic functions-i.e., functions f such that d f= 0. The inner product on the Hilbert space is (fl g) cc J d’seC’@f(z, 5) g(z, 5).
(4.12)
where the overall normalization is chosen so that (111) = 1. With the Kahler potential @ a function of Iz(‘, fi, given as in (4.1 l), commutes with J. Thus, as before, we may seek simultaneous eigenstates of the form $ = z’P( I:( ‘). The energy eigenstate condition equation for P(X),
(4.13)
for I,$ then becomes the ordinary
differential (4.14)
where x reduces Taking equation
= JzI’ and prime ’ means d/dx. Taking @(lzl’) = IzI’, as in (4.4a), this to the Laguerre differential equation (2.13) of the Landau problem. @()~(~)=2gln(l+ 1z12/2g), as in (4.4b), (4.14) reduces to the Jacobi (3.18) of the monopole problem after redefining the argument of P to be
Cl- l~12/%Yu + l#/k). This formulation in terms of Kahler metrics on two-dimensional surfaces suggests another model which is a different deformation of the Landau system. Choosing the Kahler potential @ to be @(lzl*)= the Hamiltonian
-2gln(l-
IzI’/2g),
(4.15)
(4.9) becomes
The subscript “HM” signifies that this is the stereographically projected Hamiltonian for charged particles in the presence of a “hyperbolic monopole.” By “hyperbolic monopole” we shall mean a magnetic field in three-dimensional Minkowski space which is everywhere orthogonal to the upper sheet (x3 9 R) of the hyperbolic surface
-(~~)~-(.x~)~+(x~)~=R~.
(4.17)
Using hyperbolic coordinates (p, z, d), .u’=psinhtcos~ .x’ = p sinh r sin 4 x3 = p cash z,
(4.18)
256
GERALD
V. DUNNE
we may represent a hyperbolic monopole (in the upper sheet with x3 30) (positive) charge g by means of the vector potential
A=_P(P+-Y3)‘p(p+x3)’ gx2 gx’ o. c J The corresponding
of
(4.19)
magnetic field is
B+,
(4.20)
P2
where fi = (sinh r cos 4, sinh z sin 4, -cash r) is the unit normal (in the Minkowski sense) to the hyperbolic surface. When the motion is restricted to the hyperbolic surface p = R (and x3 3 0), the quantum mechanical Hamiltonian is ~,,=1(~x(-id-A))~l~=R.
(4.21)
To understand the relationship between the hyperbolic monopole system and planar Landau system (and also the spherical monopole system) we transform stereographically projected coordinates. This projection maps the upper sheet of hyperboloid to a planar disc of radius &R-see Fig. 2-and is achieved by following change of variables:
the to the the
(4.22)
x3=0
FIG. 2. A cross section, at fixed polar angle (, of the stereographic projection of the point P on the surface of the upper sheet of the hyperboloid of radius of curvature R from the vertex of the lower sheet to the point P’ lying in the tangent plane to the hyperboloid at the vertex of the upper sheet. The hyperbolic surface projects to a planar disc of radius 2R, and the distance JzI is 1.~1= 2R tanh(r/2).
HILBERT
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257
Once again, the JB/2 factor is introduced to conform with the normalizations in (2.3) and (3.9). Recall that on the hyperbolic surface, p = R, the magnetic field has magnitude (4.23) B = g/R’. Thus, from (4.22) we see that (=I2 < 2BR2 = 2g
(4.24)
and hence we may consider the projected disc to be of radius A. In terms of these coordinates, the hyperbolic monopole Hamiltonian (4.21) becomes ~~~=B(-(~-~)‘?8~(,-~)(~~-id)+tlr’).
(4.25)
As in the spherical monopole case, the limit in which the radius of curvature R -+ 00 in such a way that B remains fixed may be considered simply as the dimensionless limit g + co. Then H,, clearly reduces to the Landau Hamiltonian (2.4) as g --f co. From (4.24), the disc described by the coordinates z, z covers the entire plane in the limit g-co. The Hilbert space of eigenstates is defined as in (4.10) and (4.12). Defining “reduced” wavefunctions I,&, 3) by (4.26) the inner product for the $ is
(4.27) 2g- 1 1-@ 2g-2dqdf, 4nig ( & ) L.
L+(z) = -
Whenf and g are restricted to the holomorphic functions, this is the standard inner product for the Hilbert space of holomorphic functions on the disc-see Bargmann’s treatise [25] on the representation theory of SO(2, 1). The reduced Hamiltonian operator (4.11) is
1 Au”=(1-,Z,2/2g)gHHM (4.28)
258
GERALD
V. DUNNE
Since A,, commutes with the angular momentum wavefunctions of the form
generator J = zd - 58, we seek
tj = z’P( cash t ), where cash t =
1 + Jz12/2g 1 - )z)2/2g’
(4.29)
The energy eigenvalue equation for $ then leads to the following ordinary differential equation for the function P(x): (I-x2)~+2(-g-i-(-g+I)x)~-2g(~-~)P=o.
(4.30)
Note that this is exactly the same differential equation (3.18) as appeared in the spherical monopole problem, with g replaced by -g. There is, however, a crucial difference. Here, the variable x ranges between 1 and co; in (3.18) x ranges between - 1 and + 1. Regular solutions to (4.30) are called associated Jacobi functions [ 161 and lead to wavefunctions (4.31) with energy (4.32)
The wavefunctions (4.31) are orthonormal [16] with respect to the inner product (4.27). There is also a continuous part of the spectrum-for a complete discussion see Refs. 26 and 27. It is instructive to analyze these wavefunctions in an algebraic formalism, as in Section 1I.B for the Landau system and Section 1II.B for the monopole system. For the hyperbolic monopole system the relevant algebra is the Lie algebra of SU(l, 1) (or SO(2, 1)). Indeed, defining the differential operators 1
K,=--
z2a+&a-&L
(4.33a)
& K-=-&d+
1
-zr’d-Jg/2;
(4.33b)
A K3=za-z&gg,
(4.33c)
HILBERT
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259
PARTICLES
these are generators of SU( 1, 1 ), [K,,
K-1 = -2K3
CK3,K*l==+K+. The hyperbolic monopole Hamiltonian (4.16) may be expressed in terms of the quadratic Casimir operator for SU( 1, 1 ), H
;(K,K+K-K+)-K:+g'
From (4.33) and (4.34) we see that the operators K, and K- act as raising and lowering operators (respectively) for angular momentum. The spectral problem may now be solved by first solving the linear differential condition
(4.36)
K-$=0,
whose solution t+kOis the lowest angular momentum state in a given “Landau level.” States degenerate (in energy) with $0, but with higher angular momentum, they are obtained by acting on $O with the raising operators K,. The regular solutions to (4.36) with integer angular momentum are
~o=~~(l-~)g-‘t, n=o,1,2
,....
(4.37)
In contrast to the (spherical) monopole case, here the “Landau levels’” are infinitelvv degenerate because successive applications of K, to tiO never produces zero (contrast to Eq. (3.38)).* Solving K, 41,= 0 leads to solutions which are either singular or al! have negative energy. Rewriting H HM=;(K+K--K#-l)+g'), we see that I//O (and hence the entire Landau level generated from it) has energy E=$O-(-n+g)(-n+g-
l)+g2)
(4.39) as noted in (4.32). Note that for a given g there exists some n for which the energy becomes negative. ’ Solving K, II/ = 0 leads to solutions which are either singular or all have negative energy.
260
GERALD V.DUNNE
In the limit g + co, the energy spectrum (4.39) reduces to the Landau spectrum and the resealed generators
contract to the Heisenberg generators bt, h, and 21,respectively.
V. CONCLUSIONS We have shown how to construct the complete single-particle Hilbert space for charged particles in an external, perpendicular magnetic field in the following twodimensional configurations: the planar Landau system, the spherical monopole system, and the hyperbolic monopole system. In terms of appropriately chosen stereographically projected complex coordinates the latter two systems reduce simply to the planar Landau model in the limit of infinite monopole charge (this limit corresponds to the infinite radius of curvature limit such that the magnetic field strength at the surface remains fixed). We have also shown that these systems are particular cases of the problem of particles constrained to a two-dimensional Kahler manifold, interacting with a gauge field whose field strength two-form is proportional to the surface’s Kahler form. In this sense we may think of the particles as interacting directly with the surface itself. The three cases treated in detail (the plane, the sphere, and the hyperboloid) are each special for two reasons. First, due to the symmetry of the surfaces about the x3-axis, there is a globally defined angular momentum operator J= ~8 - 58 which commutes with the Hamiltonian-this makes the diagonalization of the Hamiltonian simpler. Second, each of these surfaces has constant curvature (the plane has zero curvature, the sphere has constant positive curvature, and the hyperboloid has constant negative curvature). Correspondingly, there exists, in each case, an algebraic formulation of the spectral problem which effectively reduces it to the problem of solving a linear (rather than quadratic) differential condition. This produces a “Landau level” type structure, with degenerate (in energy) states being created by use of step operators for angular momentum. Note, however, that in the hyperbolic case the relevant symmetry algebra is noncompact and consequently there is also a continuous part in the spectrum (see Refs. 26 and 27 for details). Direct solutions of the energy eigenvalue equation may also be given in terms of Laguerre polynomials, Jacobi polynomials, and associated Jacobi functions for the planar, spherical, and hyperbolic systems, respectively. In the planar and hyperbolic systems the degeneracy of each “Landau level” is infinite, due to the infinite area of the surface; whereas, in the spherical system the degeneracy is finite, due to the finite area of the spherical surface. A common feature of all three models is the existence of a “lowest Landau level” of degenerate states of energy B/2. In each case the lowest Landau level states are holomorphic functions of the corresponding complex coordinate. This means that much of the
261
HIL~ERTSpACEF~RCHAR~ED PARTICLES
analysis of multi-particle states in the lowest Landau level of the planar system may be directly translated to the other systems simply by changing the Hilbert space measure factors. For example, the Laughlin [2] states
have analogues in the spherical geometry which were written down by Haldane in terms of “‘spinor coordinates” [3]. A more direct comparison arises in the stereographically projected coordinates, in which the Haidane states are
Note that in the spherical monopole system the lowest Landau level has finite degeneracy ( = 2g -+ 1 ), and so a wavefunction describing a tilling fraction v = 1/(2m -t 1) presumes a relation between the particle number N, the monopole charge g and v as g = $(N/v - 1).
APPENDIX:
PR~P~T~~~F
The Laguerre polynomials
LAGUERREAND JACOBIPOLYNOMIALS
LIpl’(x) satisfy the di~erential-difference
equation:
(A.1)
x $ L?‘(x) = nLff’(x) - (n + cc)LFL l(X). Tbe L::‘(x)
also satisfy the following difference relations: Lf.y ‘1 = L:‘(x)
- L;:l L(.x),
fA.2)
xL~+“(xt=(x-n)Lf:‘(xf-t(a+n)1;~~,(.~f.
(A.3)
The Jacobi polynomials
PF*p’(x) satisfy the differential-difference
equation:
(A.4)
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GERALDV.D~NE
Some authors (see, e.g., Ref. [16]) define related Jacobi polynomials
Cm+nVZp)“,“.“+“‘(x).
x(1+x)
Pi,(x)
as
(A.9)
The Pk,(x) satisfy the differential equation
= -Z(Z+ l)f$,
(A.lO)
and have the integral representation
(A.1 1)
m-f-1 du,
where the contour r in (A.1 1) is over the unit circle /WI = 1. The Jacobi functions @k=(x) are also solutions to the differential equation (A.lO), but are defined on the interval XE [l, co), rather than the interval XE [- 1, t] on which the Pf,,{x) are defined. The &,(.x) have the following integral representation Pf?,,(cosh z) =&
$r (cash i + o sinh i)“’ I-n ocoshi+sinhi
urn-‘-
’ da,
(A.12)
>
where the contour r in (A.12) is over the unit circle (~1 = 1.
I am grateful to A. Lerda and A. Perelomov for helpful discussions and comments and to the Aspen Center for Physics where this work was completed. REFERENCES 1. For an excellent
review,
see S. GIRVIN
AND R. PRANGE,
“The Quantum Hall Effect,” Springer-Verlag,
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HILBERT
SPACE
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263
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