Hall Ictp

A fuzzy description of Quantum Hall Physics

N.E.Grandi

Landau Levels  Single charged particle in an external uniform magnetic field B

1 2

1 2

Landau Levels  The classical dynamics is very simple B

The trajectories are circular orbits with a fixed angular velocity

Landau Levels  The quantum problem is more interesting B

Independent of χ, that can be used to label the degeneracy

Landau Levels  And we find the Landau levels B

Landau Levels  Projecting the Hilbert space into the LLL B The projection induces noncommutativity (Fuzzy plane)

If ~eB>>m we can project the space into its fundamental subspace

Landau Levels  Projecting before quantization is equivalent B

Dirac Brackets

The projection induces noncommutativity (Fuzzy plane again)

Second Class Constraint

Landau Levels  Space-space uncertainty relation: fuzzyness B The particle behaves as an incompressible droplet The degeneracy labeled by the eigenvalues of x1 is given in terms of the area per particle The filling fraction is quantized in integer multiples of e/~

Landau Levels  Some additional information B

• The integer quantum Hall effect is explained in terms of multiparticle wave functions of non interacting particles. • The quantization of the filling fraction and the observed independence of the peculiarities of the sample and stability against perturbations and disorder, is explained in terms of gauge invariance (Laughlin). • Inclusion of the crystalline structure in a magnetic field can be formulated with the help of noncommutative geometry and magnetic translations (Bedllisard, van Elst, Shultz-Baldes) The same is true for disordered lattices.

Matrix description  Successes and limitations  Integer quantum Hall effect is completely explained in the above picture. The quantization of the filling fraction is a strong topological effect originated on gauge invariance

 To study more general filling fractions, we need to introduce the notion of composite fermions or bosons and singular gauge transformations.

 For the fractions of the formν =1/n Laughlin built a complete set of wave functions on variational grounds. They take the form

Fluid dynamical description  System of many particles in a magnetic field B

Re-labeling symmetry (permutation of the particles)

Fluid dynamical description  We take a continuum limit in a macroscopic scale B Lagrange description of a fluid

Fluid dynamical description  The permutation symmetry becomes a gauge symmetry under APD B

Fluid dynamical description  The resulting theory has a very simple dynamics B

Incompresible fluid Propagation, if any, localizes at the boundary. Vortex solution

Vortices need sources

Fluid dynamical description  After quantization we get more information B

Quantum noncommutativity

There is no quantization of Quantization of the vortex the filling fraction charge in units of ν

Vortices acquire fractional statistics

Fluid dynamical description  Some additional information B

• The theory is equivalent to a U(1) ChernSimons, under the map xa = ya + (1/2πρo) εab Ab (Susskind-Bahcall)

• After solution of the constraint, the action becomes that of a chiral boson at the boundary, (Wen) •The quantization of the filling fraction is not needed to have gauge invariance (Polychronakos)

Fluid dynamical description  Successes and limitations  It has vortex (quasihole and quasiparticle) solutions.  The charge of the vortex is quantized in units of the filling fraction.

 The vortices have fractional statistics.  The dynamics is localized on the boundary  We still need to add external sources.  The filling fraction is NOT quantized!

Noncommutative description  Can we map the constraint to a commutator like [xa,xb]? B Unique way Weyl of doing that respecting map associativity (Moyal product)

Noncommutative description  We have an auxiliary Hilbert space (it is not the space of states) B

Operators in some auxiliary Hilbert space A classical state corresponds to a choice of xa operators Its off-diagonal elements represent diagonal elements represent the “mixing” the particles positionsbetween of each individual particle in this classical state

Is only consistent with an 1 dimensional space, then we have infinite number of particles

Noncommutative description  The gauge (permutation) symmetry is now given by unitary conjugations B

The unitary transformations represent re-shuffling of the particles The value of the charge is fixed by the constraint

The action is invariant up to a total trace

Noncommutative description  The dynamics will necessarily be very simple (infinite system, no boundary, no dynamics) B Particles A continuum are homogeneously of x1 distributed x1 complete and eigenvaluesonand delocalization complete delocalized on x2 on x2

Heisenberg algebra

Noncommutative description  This state is unique up to unitary transformations B The mostare localized particle Particles localized at Equivalent representation has a nonzero radius, equally separated radius of the same solution while the outmost one is at and completely delocalized infinity in the angles

Heisenberg algebra

Noncommutative description  Vortex solutions need external sources B

minimum

The most local operator, implies

We add a δ source

Noncommutative description  To have a gauge invariant functional integral, the filling fraction must be quantized B

The variation of the action must be an integer multiple of 2π The fillingWinding fraction is topologically quantized number

Noncommutative description  Quantization renders noncommutative the matrix elements of the operators B

Dirac Quantum Brackets noncommutativity Second Class Constraints

No dynamics Constraint on Classical physical states noncommutativity

Noncommutative description  Quantization renders noncommutative the matrix elements of the operators B

Generator of unitary Laughlin connection betweenconjugations

filling fraction and statistics Finite Permutation unitaryof particles conjugation Statistics related to filling fraction No dynamics Constraint on physical states

Noncommutative description  Some additional information B

• The theory is equivalent to a noncommutative U(1) Chern-Simons theory in the infinite plane, under the map xa = ya + (1/2πρo) εab Ab • There is no consistent way to formulate this theory in a bounded region of space, there is no local chiral boson (Grandi-Silva, Lugo, Balanchadran-Gupta-Kurkcouglu) • In R^2 the theory is classically (GS) and quantum mechanically (Kaminsky-Okawa-Ooguri) equivalent to the commutative U(1) Chern-Simons • Nevertheless the quantization of the filling fraction survives to this equivalence (Polychronakos)

Noncommutative description  Successes and limitations  It has vortex (quasihole and quasiparticle) solutions.  The charge of the vortex is quantized in units of the filling fraction.  The vortices have fractional statistics.

 The particles have statistics according to the filling fraction.  The filling fraction IS quantized!  We still need to add external sources.  We’ve lost boundary dynamics (infinite system).

Matrix description  We want a finite dimensional auxiliary Hilbert space B

Operators in some auxiliary Hilbert space Its diagonal elements represent the positions, off diagonal elements represent mixing Additional “boundary” degrees of freedom (vectors in the fundamental)

Is only consistent with an 1 dimensional We get a modified action space, then we have infinite number of and constraint, that allow particles for finite dimension

Matrix description  Unitary (permutation) symmetry still present B

Unitary re-shuffling of the particles acts on ψ in the fundamental

The generator has a fixed value

Matrix description  The dynamics now much more interesting B

Most general solution, xp and yp are integration constants

Matrix description  We have a solution representing a Hall droplet B

The Particles most are outer localized particleathas fixed a finite radiusradius and completely = quantum hall droplet delocalized around the circle

Matrix description  Vortex solutions don’t need any external source B The vortex pushes the boundary away a finite distance

We don’t add any δ source

Matrix description  New solutions are obtained representing edge states B

This is a new solution, not unitarly equivalent to the previous Diagonal radius states are mapped into nondiagonal ones

Matrix description  The quantum theory is constructed as before B

The have We fillingthe fraction sameiscanonical topologically a commutators quantized exactly for xas before And additional canonical commutators for ψ The constraint relates statistics of physical states with ν

Matrix description  The states can be found explicitly in terms of a creation and annihilation basis B

This space of solutions is isomorphic to Laughlin wave functions!

Matrix description  Some additional information B

• The theory is equivalent to a matrix U(1) ChernSimons theory. • It can also be mapped into the Calogero or Calogero-Suterland models (Polychronakos) • Its quantum states have been shown to be exactly given by the Laughlin wave functions (Hellerman-Van Raamsdonk, Cappelli-Riccardi) •It can be extended to include multilayer system and spin of the fundamental constituent (PolychronakosMorariu)

•It is not known how to use this model to describe more general filling fractions p/q

Matrix description  Successes and limitations  This theory can describe finite samples.  It has vortex (quasihole and quasiparticle) solutions included in the theory without adding any external source.  The charge of the vortex is quantized in units of the filling fraction.  The vortices have fractional statistics.

   

The particles have statistics according to the filling fraction. The filling fraction is quantized as 1/n We have perturbations describing edge states The states of the theory are Laughlin wave functions!

 It does not apply to more general filling fractions

Conclusions and outlook  What’s left?  The Chern-Simons Matrix Model solves many of the problems of the previous formulations of FQHE.

 It captures much of the physics of this system, including

topological quantization, edge states and fractional statistics of its excitations, its quantum states being described by Laughlin wave functions

 Some more elaborated test still to be passed. For example the calculation of tunneling exponents, the phase transition to a Wigner crystal and the possibility of nonhomogeneous phases

 It still to be understood how to formulate a matrix model useful

to describe more general fillings. Moreover, the definition of local observables is always difficult in noncommutative theories, so they should be translated to fuzzy analogs.

Thanks!