ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 2, с. 373-379
ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ
SUPER-LANDAU MODELS: UNITARITY AND HIDDEN SYMMETRIES
©2010 E. A. Ivanov*
Bogoliubov Laboratory of Theoretical Physics, JINR, Dubna, Russia Received April 17,2009
We review a recent progress in constructing and studying superextensions of the Landau problem of a quantum particle moving on the two-sphere S2. These are the superspherical and superflag Landau models describing a nonrelativistic particle on the supermanifolds SU(2\1)/U(1|1) and SU(2\1)/[U(1) x U(1)]. Though in both models, under the "naive" definition of the inner product, there are negative-norm states, all norms can be made positive-definite by introducing a nontrivial "metric" in the space of states. So both models are unitary. The superflag model is shown to be quantum-equivalent to the supersphere model for the special choice of the intrinsic parameters. We also address the planar limit of the superspherical model, in which there arises the hidden world-line N = 2 supersymmetry. An analog of this phenomenon in the superflag model is a dynamical enhancement of its SU(2\ 1) symmetry to SU(2\2).
1. INTRODUCTION
The Landau model [1] describes a charged particle moving on a plane through which a constant uniform magnetic field passes. A spherical Landau model [2] describes a charged particle on a 2-sphere 52 - SU(2)/U(1) in the background of the Dirac monopole placed in the centre of S2. These models constitute a theoretical basis of the Quantum Hall Effect (QHE)[3].
The aim of the present talk is to outline the characteristic features of two minimal superextensions of the S2 model: (i) Landau problem on the (2 + + 2)-dimensional supersphere SU(2|1)/U(111) [4,5]; (ii) Landau problem on the (2 + 4)-dimensional superflag SU(2|1)/[U(1) x U(1)] [5, 6]. Being associated with the same supergroup, these two quantum-mechanical models are closely related to each other, and this relation will be explained here from the point of view of the corresponding coset superspace geometries. Also, their common feature (which manifests itself already in some toy fermionic Landau-type problems) is that under the standard ("naive") definition of the inner product of functions on these coset manifolds, there are quantum states with negative norms. This unpleasant feature can be "cured" by introducing a nontrivial operator metric on the space of quantum states ("the Hilbert space"), and we give its explicit form1).
E-mail: eivanov@theor.jinr.ru
1)1 Some other versions of supersymmetric Landau problem associated with the supergroups different from SV(2|1) were treated in [7].
The large S2 radius limits of the S2 super Landau problems (planar super Landau models) were considered in [8—10] (see also [11]). One of such planar models is exposed in Section 5 as a simplified example sharing most of the peculiarities of the original super S2 models.
2. PRELIMINARIES 2.1. Planar Bosonic Landau Model
The bosonic Landau model is described by the following Lagrangian and Hamiltonian:
Lb = lz|2 — iK (ZI — zz) = (1)
= z2 + (Az z + Az I),
Az = —iKZ, Az = iKZ,
Hb = a^a + k, a = i(dz + kz ), (2)
a = i(dz — kz), [a, a^] = 2k.
The second term in (1) is the simplest example of d = 1 Wess—Zumino (WZ) term. The invariances of this model are "magnetic translations" and 2D target SO(2) - U(1) rotations:
Pz = —i(dz + k I), Pz = —i(dz — kz ), Fb = zdz — Idz, [Pz, Pz] = 2k, [H, Pz] = [H, Pz] = [H, Fb] = 0.
2.2. Wave Functions
The lowest Landau level (LLL) is defined by the eigenvalue condition
H tf (0) = k* (0)
which is solved by
аФ(о)^, z) = 0 д + nz)^(o) =
(0)
= e
-k\z\2
Ф(о)(z),
where 0)(z) is a holomorphic function. On the tth Landau level (LL) wave function ^(£), the Hamilto-nian H takes the following eigenvalue
H Ф
к(2£ + 1)Ф(,).
This wave function is also reduced to a holomorphic function
(z,z) = (a^)e *$)(z,z) = = m — kz)]£e-^^(z).
Each LL is infinitely degenerate due to the (Pz,Pe) invariance.
2.3. Generalization to S2 - SU (2)/U (1) An S2 analog of the planar Lagrangian Lb reads
Lb =
(1 + |z|2)2
| Z |2 + is
1 + lzl
(Zz — zzz). (3)
Lf = а — îk(cz+а),
(4)
Hf = — or а — к,
a = d(- — k Z, ajf = dz — k (. The corresponding invariances are generated by:
n = dz + k z, n = de + k Z, (5) Ff = Z dc — Zde,
[Hf, n ] = [Hf, ne] = [Hf ,Ff ] = o.
The space of states consists of the ground state and single excited state:
^(0) = e-KZZ^0 ( Z), ^(1) = eKZZ^1 {Z), (6)
a^(0) = at ^(1) =0, ^o = Ao + Z Bo,
= A1 + ZB1, H^(0) = —k^(0) , H^(1) = K^(1) .
The natural supertranslation invariant choice of the inner product is
(7)
The 2nd term is the SU(2)/U(1) WZ term; for simplicity, hereafter we choose the S2 radius equal to 1. The Ll wave functions for t = 0,t = 1,... are finite-dimensional SU(2) irreps, with the "spins" s,s + + 1, s + 2,... They are represented by polynomials of z which are square-integrable on S2; the invariance of (3) and of the corresponding quantum Hamiltonian is SU(2). The transformations leaving (3) invariant (modulo a total t derivative) are the standard SU(2) ones realized as motions of the 2-sphere S2 in the holomorphic CP1 parametrization
5z = a + iuz + az"2.
The degeneration of LL is finite because SU(2) is compact as opposed to its contraction, the magnetic translation group of the planar limit, which is non-compact.
2.4. ToyFermionic "Landau Model"
The Lagrangian and Hamiltonian of this model are literal fermionic counterparts of (1), (2):
^ = j wat ^{z,z), (^(0)|^(1)) = 0, ) = 2kaO aO + B0B0, (^(1) |^(1)) = -2ka1a1 — B1B1.
Already in this simplest version of fermionic Landau model we observe the presence of states with negative norms. However, this unwanted feature can be removed by redefining the inner product as follows:
((tty)) := (Gtty), G (>0) + ^(1)) = ^(0) — ^(1), G = —k-1 Hf.
With respect to the new product all norms are positive.
All symmetry generators (5) commute with the metric G, so the new inner product is still invariant. However, the hermitian conjugation properties of the operators which do not commute with G, change. Consider the generic case, not necessarily restricted to the specific model under consideration [9]. Let O be some operator, such that [H, O] = 0. Then O$ = = GOtG = Ot + GOG, Og = [G, O], and [H, OG] = = 0. The symmetry generators that do not commute with G thus can generate "hidden" symmetries of the relevant model. Examples will be given below.
3. SUPERSPHERICAL LANDAU MODEL 3.1. Definitions
The supergroup SU(2\1) is the minimal rank 2 semi-simple supergroup extending U(2) ~ (F, J(ik)) by the U(2) doublet of fermionic charges (Qi, Qk):
{Qi,Qk } = eikF + Jik, (8)
{Qi,Qk} = {Qi,Qk} = 0.
TheRiemann supersphere CP (111) = SU(2\1)/U (1 \ 1), where U(1 \ 1) ~ (J3,F,Q2,Q2) is a complex super-manifold
Z^ = (Z0,Z1) = (z,z), ZB = (Z0,Z1) = (z,a),
1
1
2
where z is a complex coordinate of CP1 ~ ~SU(2)/U(1) and Z is its anticommuting partner. Their SU(2\1) transformations are analytic
5z = iXz + e + ez2 — (e2 + ze1) Z, (9)
St = ^(А + Ж + е1 +
A, e, e, ц being infinitesimal U(2) parameters and e% Grassmann parameters.
The supersphere (SS) is the natural superextension of the customary sphere S2 ~ SU(2)/U(1). It is a Kahler supermanifold, with the Kahler 2-form
F = 2idZA Л dZBdBOaK = dA.
Here, K = log (1 + zz + ZZ) is the Ka hler potential and
A = —i (dZa0a — dZЁd^j K = dZaAa + dZвАЁ is the Kaahler connection.
3.2. CP(11 Model: Lagrangian and Hamiltonian
The invariant Lagrangian of the Landau model on CP(1'1), a natural extension of the S2 model, reads:
L
9zz =
1 + CC
9(z
(1 + zz + CC)
zz
(1 + zz)2'
2 ' 3zZ =
9<C
zZ
(1 + zz)2' 1
V(fN= 0 ^ Ф0Т) = e"^[A0(z) + Z^o(z)],
,(N )
H Ф,
(N )
0.
For all t > 1 the wave functions are defined by:
Ф
(N )
Ф
(N) (-)i
^ VN+1) . . . V(N+2p-1^ • V(N+2—) ' z С z
Lp=I
(±)- -N^ (z,Z)
Ф
V(N) Ф± =0 ^ Ф± = e-N(±
A (N)
375
(14)
(-) i ,
(15)
H Ф^) = £(£ + 2N )Ф
The natural definition of the SU(2|1) invariant inner product is
= y d^oe-K^ll^ll2 = (16)
= d^0e-K d^0 = dzdzdz dz.
The wave functions ^ for different t are orthogonal to each other with respect to this product. The norm at fixed t is expressed through the fields in the Z expansions
^ = At + Z^e and = xt + ZF
r dzdz
as
11Ф
iN )ll2
(1 + zz)2(N+i)+1
x
(17)
ZAZÈgBA + n(ZaAA + ZBAB), (10)
1 + zz
The quantum Hamiltonian is given by the expression H = - (-1)a(a+b) gASV(N\ (11)
gAegcB = , vAN) = dA - N (dAK), vB) = dB + N (d bK).
Here, a, b are Grassmann parities associated with the indices A and B.
The £ = 0 (i.e., LLL) wave function ) (Z, Z) is covariantly analytic:
- £ (2N + £) |Ai|2 - £фф - £(Xi + z$i) x
. . 2 (N + £) + 1 . l2 x (xe + ф) + 1 + z-z-xexe + wa
Thus we observe that the "natural" norm is not positive definite like in the toy fermionic Landau model, and the associated quantum theory can break unitarity. However, there is an alternative definition of the SU(2\1) invariant norm, such that the latter is positive-definite. We shall show this in the context of the more general (superflag) model.
4. SUPERFLAG LANDAU MODEL 4.1. Geometry
The (1\2)-dimensional superflag (SF) is the coset superspace SU(2\1)/[U(1) x U(1)]:
M = (z,Z,0, Zm = (z,Z,i),
z 1
(12)
ф(Х = v N+1) • • • v (N+2i-1^, (13)
Like the supersphere, the superflag is a Kai hler super-manifold:
F1 = 2idZ Л dZdd log K = dB, F2 = -2idZ Л dZdd log K2 = dA,
r^j
x
o
B = i [dZM дм - dZ MdM) log Кг =
= dZM Bm + dZM BM, A = -i (dZмдм - dZMdM) log К = = dZM Am + dZM AM. Here Кг = 1 + (Z + z£) (( + z() +
These conditions amount to the covariant chirality constraints:
'M ) = D- Ф PN 'M ) =0,
ph
{D±, = ±D
±±
[D--, D++] = 2J3 = 2N', {D+, D-} = {D-, D+} = 0.
Here, D±±, D±, D± are the proper covariant derivatives on the supermanifold SU(2^)/[U(1) x U(1)]
with the U(1) x U(1) connections related to the maThe invariant Lagrangian of the simplest variant of , . , f , f f IU тт,лЛ тт,лЛ , ^ CI-.T , j r • • и tri
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