ЯДЕРНАЯ ФИЗИКА, 2012, том 75, № 10, с. 1298-1304




©2012 E. A. Ivanov*

Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna, Russia

Received October 13, 2011

It is a brief review of a new class of N = 2 supersymmetric Landau models which generalize the superplane Landau model by extending it to an arbitrary magnetic field on any two-dimensional manifold M2. Using an off-shell N = 2 superfield formalism, it is shown that these models are characterized by two independent potentials given on M2. The relevant Hamiltonians are factorizable and in the special case, when both the Gauss curvature and the magnetic field are constant over M2, admit infinite series of factorization chains, which implies the integrability of the associated systems. For the particular model with the CP1 bosonic manifold, the spectrum and eigenvectors are explicitly given.


The renowned Landau model [1] describes a charged particle moving on a plane orthogonal to a constant uniform magnetic flux. A spherical generalization of this model was constructed by Hal-dane [2]. It describes a charged particle on the 2-sphere S2 ~ SU(2)/U(1) in the background of Dirac monopole placed in the center. The Landau-type models have plenty of applications. In particular, they provide a theoretical basis of the Quantum Hall Effect (QHE) [3].

By definition, superextended Landau models are models of non-relativistic particles moving on supergroup manifolds with S2 or its planar limit as a "body". Minimal superextensions of the S2 Haldane model were constructed in [4, 5]. They include:

(i) Landau model on (2|2)-dimensional super-sphere SU(2|1)/U(1| 1) [5];

(ii) Landau model on (2|4)-dimensional superflag SU(2|1)/[U(1) x U(1)] [4,5].

The large S2 radius limits of these systems yield the planar super-Landau models. They were introduced and studied in [6, 7]!).

The most interesting feature of the super-planar Landau models is their hidden worldline N = 2 supersymmetry. One starts with a model invariant under some target supersymmetry and finally finds the N = 2, d = 1 supercharges which square on the Hamiltonian of the system. Thus, the super-planar

E-mail: ivanov@theor.jinr.ru

!) See also [8] where an alternative planar super-Landau model based on the contraction of the supergroup OSp(1\2) was considered.

Landau models simultaneously provide a class of the supersymmetric Quantum Mechanics (SQM) models. SQM models [9] have a lot of applications.

Based on this notable property, a natural extension of the super-planar Landau models can be constructed as follows. One takes the worldline N = = 2 supersymmetry as the primary principle and constructs the most general N = 2 SQM model which involves the standard superplane Landau model [6] as a particular case. Such a construction has been recently accomplished in [ 10], based on N = 2, d =1 superfield formalism. In this contribution, we present a survey of this new class of super-Landau models.


Before turning to the main subject, let us remind the salient features of the bosonic Landau-type models and their superextensions.

2.1. Planar Bosonic Landau Model

The customary Landau model is described by the following Lagrangian and Hamiltonian:

Lb = ^|2 - in (zz — zz) = (2.1)

= z|2 + (Azz + Anz),

Az = —iK z, A? = iKz, dzAz — dz Az = —2iK, ^ aa0=^K}


a = i(dz + kz ), a^ = i(dz — kz), (2.3)

[a, a)] = 2k.

The invariances of this model are "magnetic translations" and 2D rotations generated by the following operators:

Pz = -i(dz + kz), P-z = -i(d-z - kz), (2.4) Fb = zdz - Zdz, [Pz,Pz]=2K, [H,Pz ] = [H,Pz] = [H,Fb] = 0.

The full set of wave functions corresponding to different Landau Levels (LL) is as follows:

(i) Lowest Landau level (LLL), H= k^(o) :

a^{o)(z, z) = 0 & (dz + kz)#(o) =

= 0 - ^(o) = e-Klzl

Ao) (z),

(ii) nth excited LL:

yH(z,z) = [i(dz - KZ)]ne-Klzl2^(n)(z), H ty(n) = K(2n + 1)tf(n).

Each LL is infinitely degenerate due to (Pz,Pz) invariance. The wave functions form infinite-dimensional unitary irreps of this non-compact group, with the basis consisting of the monomials zm, m > 0. They possess invariant norms:

\№(n) ||2 -

- J dzdze-^ïi^iz^^iz) < oo

for any monomial n) (z) — zm.

2.2. Generalization to S2 An S2 analog of the planar Lagrangian Lb is


Lb =

+ is


(1 + r2lzl2 )2

Iz I2 +



1 + r2IzI2

(Zz — Zz).

The second term is the d = 1 WZ term on the coset SU(2)/U(1), r being the "inverse" radius of S2. The first term is the d = 1 pullback of the S2 interval.

The wave functions in this case are finite-dimensional SU(2) irreps, s,s + 1,s + 2, ... being their "spins". The LLL wave function is determined by the covariant analyticity condition on S2

V, *«,) = 0, (2.7)

V, = (1 + r2\z\2)d, + U(1) connection.

Each LL is finitely degenerated since the wave functions are SU(2) irreps. The limit r — 0 yields the planar Landau model.


3.1. Worldline Supersymmetryvs. Target—Space Supersymmetry

Super-Landau models are quantum-mechanical models for a charged particle on a homogeneous supermanifold, such that the "bosonic" truncation is either Landau's original model for a charged particle on a plane or Haldane's spherical version of it. There are two approaches to constructing such extensions.

A. Worldline supersymmetry:

t ^ (t, 0,0), z,z ^ Z(t, 0, 0), Z(t, 0, 0),

z,z ^ (z, z, ^,tp,...) — worldline supermultiplet.

This option yields a version of SQM.

B. Target—space supersymmetry:

group manifold : (z, z) ^ ^ supergroup manifold : (z, z, Z, Z)

(Pz, Pz, Fb, k) ^ (PZ,P-Z, nc, n-c,Fb,Ff ,k,...),

nc = dc + kz, n, = d, + kz, Ff = Zdc - Zd,, {nc, n,} = 2k.

The geometrical meaning of this procedure in the simplest case is that 2-dimensional plane (z, z) is extended to a (2\2)-dimensional superplane (z, z, Z, Z), where Z, Z are new complex fermionic coordinates.

3.2. Superplane Landau Model

Planar super-Landau models are the large radius limits (contractions) of the supersphere and superflag Landau models. One makes explicit the S2 radius R, properly rescales Hamiltonians, and sends R — to. The supersphere SU(2\1)/U(1\1) goes into an (2\2)-dimensional superplane.

The superplane Landau model is determined by the following Lagrangian and Hamiltonian:

L = Lf + Lb = \z\2 + ZC - (3.1) - iK[zz - zz + ZZ + ZZ

H = o^o — at a = d- d- — dzdz + (3.2)

+ K (zdz + Zdz - Zdz - Zdc) + K2 (zz + ZZ) ,

where the operators a, at were defined in (2.3) and

a = d- — kZ, aJf = d- — kZ.


The invariances are generated by Pz, Pz, n, n and by the new spinorial generators

Q = zdz — Zd-z, QÏ = Zd- + Zdz,


together with

C = zdz + Zdz - zdz - (d^.

They generate the supergroup ISU(1|1), contraction of SU(2|1):

{Q,Q^} = C, [Q,Pz ]= inc, (3.5)

Q Пс} = iPz.

3.3. Norms and Hidden Worldline Supersymmetry

The natural ISU(1|1)-invariant inner product is defined as

(Ф\Ф) = J d/лф (z, z] C, ()ip (z, z] С, C) , (3.6)

d^ = dzdzd(d(.

This definition leads to negative norms for some component wave functions. To make all norms not negative we need to introduce the "metric" operator: 1

and so it is a singlet of N = 2 supersymmetry. Hence N = 2 supersymmetry is unbroken and all higher LL form irreps of it.

3.4. Superfield Formulation

Superfield formulation of the superplane model was given in [11]. It makes manifest the hidden N = 2 supersymmetry of this model.

This formulation makes use of N = 2, d = 1 superspace in the left-chiral basis (т = t + i99,9,9). The basic objects are N = 2, d = 1 chiral bosonic and fermionic superfields Ф = z(t) + 9х(т), Ф = ф(т) + + 9Ь(т), with х(т) and Н(т) being auxiliary fields:

DФ = DФ = 0.

The superfield action yielding the superplane model action is:

S = / dtd29l ФФ + ФФ +


G = - [9Ç9ç- + k2CC + k(C9ç-C9ç-)] , (3.7) + p [ФДФ - ФДФ] }, p = l/(2v^).

((^)) dp (G№.

The full Hamiltonian H commutes with G, so H = = Ht = H*, where \ denotes Hermitian conjugation with respect to the new inner product. However, the Hermitian conjugation properties of the operators, which do not commute with G, change.

Let O be generator of some symmetry, such that [H, O] = 0. Then

O* = GO^G = Of + GOfG, Og = [G, O], (3.8)

and OG is another operator such that [H, OG] = = 0. The symmetry generators that do not commute with G thus generate, in general, additional "hidden" symmetries.

In our case G commutes with all ISU(1|1) generators, except for Q, Qt. Thus, the conjugation rules of the latter change:

The auxiliary fields h and % are eliminated by their algebraic equations of motion as % = 2ipZ, h = = —2ipz. Then the action written in terms of physical fields reads

S dt


zz - zz + zz - Zz + (3.12)

+ zz + cc

Q* = Qf - -S,


S = o^a = i (dzd(- + k2z( — Kzd,= — n(dz) , S* = aa*,

where a* = —a^. The operators S, S*, H can be checked to form N = 2,d = 1 superalgebra:

{S,S *} = 2kh, {S,S} = {S*,S*} = 0, (3.10)

[H, S] = [H, S*] = 0.

The LLL ground state is annihilated by S, S*

= S*^(0) = 0,

The natural idea was to construct a generalized N = 2 supersymmetric Landau model by passing to the most general N = 2 superfield action, S ^ Sgen. It was recently accomplished in [ 10].


4.1. Most General N = 2 Superfield Action

A generalization of the superplane model super-field action (3.11) is as follows:

Sgen = У dtd29^K (Ф, Ф) + У(Ф, Ф)ФФ + (4.1)

+ p (ФDФ - ФD^) } = ^ dtL.

It involves two independent superfield potentials, K(Ф,Ф), V(Ф,Ф), and becomes the superplane model action in the flat limit, when K ^ ФФ, V ^ 1.

After eliminating auxiliary fields in Ф = z + ..., Ф = ф + ..., and setting 4p2 = 1, the component Lagrangian reads

Lcomp = V-1Zф + i (ZKz - ФKz) + (4.2)

+ ф — terms.

By introducing the notation ZA = (г,ф), it can be rewritten as

L = ZAZÈgBA + ZaAa + Zb Ab


where, e.g.,

w - Vz Vz




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