A fuzzy description of Quantum Hall Physics N.E.Grandi
<ul><li>Single charged particle in an external uniform magnetic field </li></ul>Landau Levels 1 1 2 B 2
Landau Levels <ul><li>The classical dynamics is very simple </li></ul>B The trajectories are circular orbits with a fixed ...
Landau Levels <ul><li>The quantum problem is more interesting </li></ul>B Independent of   , that can be used to label th...
Landau Levels <ul><li>And we find the Landau levels </li></ul>B
Landau Levels <ul><li>Projecting the Hilbert space into the LLL </li></ul>The projection induces noncommutativity (Fuzzy p...
Landau Levels Dirac Brackets <ul><li>Projecting before quantization is equivalent </li></ul>B Second Class Constraint The ...
Landau Levels <ul><li>Space-space uncertainty relation: fuzzyness  </li></ul>B The particle behaves as an incompressible d...
Landau Levels <ul><li>Some additional information </li></ul>B <ul><li>The integer quantum Hall effect is explained in term...
Matrix description <ul><li>Integer quantum Hall effect is completely explained in the above picture. The quantization of t...
<ul><li>System of many particles in a magnetic field </li></ul>Fluid dynamical description B Re-labeling symmetry (permuta...
Fluid dynamical description <ul><li>We take a continuum limit in a macroscopic scale </li></ul>B Lagrange description of  ...
Fluid dynamical description <ul><li>The permutation symmetry becomes a gauge symmetry under APD </li></ul>B
Fluid dynamical description <ul><li>The resulting theory has a very simple dynamics </li></ul>B Incompresible fluid Vortic...
Fluid dynamical description <ul><li>After quantization we get more information </li></ul>B Quantization of the vortex char...
Fluid dynamical description <ul><li>Some additional information </li></ul>B <ul><li>The theory  is equivalent to a U(1) Ch...
Fluid dynamical description <ul><li>It has vortex (quasihole and quasiparticle) solutions. </li></ul><ul><li>The charge of...
Noncommutative description <ul><li>Can we map the constraint to a commutator like  [x a ,x b ]? </li></ul>B Unique way of ...
Noncommutative description <ul><li>We have an auxiliary Hilbert space (it is not the space of states) </li></ul>B Operator...
Noncommutative description <ul><li>The gauge (permutation) symmetry is now given by unitary conjugations </li></ul>B The v...
Noncommutative description <ul><li>The dynamics will necessarily be very simple (infinite system, no boundary, no dynamics...
Noncommutative description <ul><li>This state is unique up to unitary transformations </li></ul>B Heisenberg algebra Equiv...
Noncommutative description <ul><li>Vortex solutions need external sources </li></ul>B We add a    source The most local o...
Noncommutative description <ul><li>To have a gauge invariant functional integral, the filling fraction must be quantized <...
Noncommutative description Constraint on physical states <ul><li>Quantization renders noncommutative the matrix elements o...
Noncommutative description Permutation of particles Finite unitary conjugation <ul><li>Quantization renders noncommutative...
Noncommutative description <ul><li>Some additional information </li></ul>B <ul><li>The theory  is equivalent to a noncommu...
Noncommutative description <ul><li>It has vortex (quasihole and quasiparticle) solutions. </li></ul><ul><li>The charge of ...
Matrix description <ul><li>We want a finite dimensional auxiliary Hilbert space </li></ul>B Operators in some auxiliary Hi...
Matrix description <ul><li>Unitary (permutation) symmetry still present </li></ul>B Unitary re-shuffling of the particles ...
Matrix description <ul><li>The dynamics now much more interesting </li></ul>B Most general solution, x p  and y p  are int...
Matrix description The most outer particle has a finite radius = quantum hall droplet <ul><li>We have a solution represent...
Matrix description <ul><li>Vortex solutions don’t need any external source </li></ul>B We don’t add any    source The vor...
Matrix description <ul><li>New solutions are obtained representing edge states </li></ul>B This  is a new solution, not un...
Matrix description We have the same canonical commutators for x a <ul><li>The quantum theory is constructed as before </li...
Matrix description <ul><li>The states can be found explicitly in terms of a creation and annihilation basis </li></ul>B Th...
Matrix description <ul><li>Some additional information </li></ul>B <ul><li>The theory is equivalent to a matrix U(1) Chern...
Matrix description <ul><li>This theory can describe finite samples. </li></ul><ul><li>It has vortex (quasihole and quasipa...
Conclusions and outlook <ul><li>The Chern-Simons Matrix Model solves many of the problems of the previous formulations of ...
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A Fuzzy Description of Quantum Hall Physics

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A Fuzzy Description of Quantum Hall Physics

  1. 1. A fuzzy description of Quantum Hall Physics N.E.Grandi
  2. 2. <ul><li>Single charged particle in an external uniform magnetic field </li></ul>Landau Levels 1 1 2 B 2
  3. 3. Landau Levels <ul><li>The classical dynamics is very simple </li></ul>B The trajectories are circular orbits with a fixed angular velocity
  4. 4. Landau Levels <ul><li>The quantum problem is more interesting </li></ul>B Independent of  , that can be used to label the degeneracy
  5. 5. Landau Levels <ul><li>And we find the Landau levels </li></ul>B
  6. 6. Landau Levels <ul><li>Projecting the Hilbert space into the LLL </li></ul>The projection induces noncommutativity (Fuzzy plane) B If ~ eB >> m we can project the space into its fundamental subspace
  7. 7. Landau Levels Dirac Brackets <ul><li>Projecting before quantization is equivalent </li></ul>B Second Class Constraint The projection induces noncommutativity (Fuzzy plane again)
  8. 8. Landau Levels <ul><li>Space-space uncertainty relation: fuzzyness </li></ul>B The particle behaves as an incompressible droplet The degeneracy labeled by the eigenvalues of x 1 is given in terms of the area per particle The filling fraction is quantized in integer multiples of e/ ~
  9. 9. Landau Levels <ul><li>Some additional information </li></ul>B <ul><li>The integer quantum Hall effect is explained in terms of multiparticle wave functions of non interacting particles. </li></ul><ul><li>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). </li></ul><ul><li>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. </li></ul>
  10. 10. Matrix description <ul><li>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 </li></ul><ul><li>To study more general filling fractions, we need to introduce the notion of composite fermions or bosons and singular gauge transformations. </li></ul><ul><li>For the fractions of the form  =1/n Laughlin built a complete set of wave functions on variational grounds. They take the form </li></ul><ul><li>Successes and limitations </li></ul>
  11. 11. <ul><li>System of many particles in a magnetic field </li></ul>Fluid dynamical description B Re-labeling symmetry (permutation of the particles)
  12. 12. Fluid dynamical description <ul><li>We take a continuum limit in a macroscopic scale </li></ul>B Lagrange description of a fluid
  13. 13. Fluid dynamical description <ul><li>The permutation symmetry becomes a gauge symmetry under APD </li></ul>B
  14. 14. Fluid dynamical description <ul><li>The resulting theory has a very simple dynamics </li></ul>B Incompresible fluid Vortices need sources Propagation, if any, localizes at the boundary. Vortex solution
  15. 15. Fluid dynamical description <ul><li>After quantization we get more information </li></ul>B Quantization of the vortex charge in units of  Vortices acquire fractional statistics There is no quantization of the filling fraction Quantum noncommutativity
  16. 16. Fluid dynamical description <ul><li>Some additional information </li></ul>B <ul><li>The theory is equivalent to a U(1) Chern-Simons, under the map x a = y a + (1/2  o )  ab A b (Susskind-Bahcall) </li></ul><ul><li>After solution of the constraint, the action becomes that of a chiral boson at the boundary, (Wen) </li></ul><ul><li>The quantization of the filling fraction is not needed to have gauge invariance (Polychronakos) </li></ul>
  17. 17. Fluid dynamical description <ul><li>It has vortex (quasihole and quasiparticle) solutions. </li></ul><ul><li>The charge of the vortex is quantized in units of the filling fraction. </li></ul><ul><li>The vortices have fractional statistics. </li></ul><ul><li>The dynamics is localized on the boundary </li></ul><ul><li>We still need to add external sources. </li></ul><ul><li>The filling fraction is NOT quantized! </li></ul><ul><li>Successes and limitations </li></ul>
  18. 18. Noncommutative description <ul><li>Can we map the constraint to a commutator like [x a ,x b ]? </li></ul>B Unique way of doing that respecting associativity Weyl map (Moyal product)
  19. 19. Noncommutative description <ul><li>We have an auxiliary Hilbert space (it is not the space of states) </li></ul>B Operators in some auxiliary Hilbert space Its diagonal elements represent the positions of each individual particle in this classical state A classical state corresponds to a choice of x a operators Is only consistent with an 1 dimensional space, then we have infinite number of particles Its off-diagonal elements represent “mixing” between the particles
  20. 20. Noncommutative description <ul><li>The gauge (permutation) symmetry is now given by unitary conjugations </li></ul>B The value of the charge is fixed by the constraint The unitary transformations represent re-shuffling of the particles The action is invariant up to a total trace
  21. 21. Noncommutative description <ul><li>The dynamics will necessarily be very simple (infinite system, no boundary, no dynamics) </li></ul>B Heisenberg algebra A continuum of x 1 eigenvalues and complete delocalization on x 2 Particles are homogeneously distributed on x 1 and complete delocalized on x 2
  22. 22. Noncommutative description <ul><li>This state is unique up to unitary transformations </li></ul>B Heisenberg algebra Equivalent representation of the same solution Particles are localized at equally separated radius and completely delocalized in the angles The most localized particle has a nonzero radius, while the outmost one is at infinity
  23. 23. Noncommutative description <ul><li>Vortex solutions need external sources </li></ul>B We add a  source The most local operator, implies minimum
  24. 24. Noncommutative description <ul><li>To have a gauge invariant functional integral, the filling fraction must be quantized </li></ul>B The variation of the action must be an integer multiple of 2  Winding number The filling fraction is topologically quantized
  25. 25. Noncommutative description Constraint on physical states <ul><li>Quantization renders noncommutative the matrix elements of the operators </li></ul>B Quantum noncommutativity Classical noncommutativity No dynamics Second Class Constraints Dirac Brackets
  26. 26. Noncommutative description Permutation of particles Finite unitary conjugation <ul><li>Quantization renders noncommutative the matrix elements of the operators </li></ul>B Constraint on physical states No dynamics Generator of unitary conjugations Statistics related to filling fraction Laughlin connection between filling fraction and statistics
  27. 27. Noncommutative description <ul><li>Some additional information </li></ul>B <ul><li>The theory is equivalent to a noncommutative U(1) Chern-Simons theory in the infinite plane, under the map x a = y a + (1/2  o )  ab A b </li></ul><ul><li>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) </li></ul><ul><li>In R^2 the theory is classically (GS) and quantum mechanically (Kaminsky-Okawa-Ooguri) equivalent to the commutative U(1) Chern-Simons </li></ul><ul><li>Nevertheless the quantization of the filling fraction survives to this equivalence (Polychronakos) </li></ul>
  28. 28. Noncommutative description <ul><li>It has vortex (quasihole and quasiparticle) solutions. </li></ul><ul><li>The charge of the vortex is quantized in units of the filling fraction. </li></ul><ul><li>The vortices have fractional statistics. </li></ul><ul><li>The particles have statistics according to the filling fraction. </li></ul><ul><li>The filling fraction IS quantized! </li></ul><ul><li>We still need to add external sources. </li></ul><ul><li>We’ve lost boundary dynamics (infinite system). </li></ul><ul><li>Successes and limitations </li></ul>
  29. 29. Matrix description <ul><li>We want a finite dimensional auxiliary Hilbert space </li></ul>B Operators in some auxiliary Hilbert space Its diagonal elements represent the positions, off diagonal elements represent mixing Is only consistent with an 1 dimensional space, then we have infinite number of particles Additional “boundary” degrees of freedom (vectors in the fundamental) We get a modified action and constraint, that allow for finite dimension
  30. 30. Matrix description <ul><li>Unitary (permutation) symmetry still present </li></ul>B Unitary re-shuffling of the particles acts on  in the fundamental The generator has a fixed value
  31. 31. Matrix description <ul><li>The dynamics now much more interesting </li></ul>B Most general solution, x p and y p are integration constants
  32. 32. Matrix description The most outer particle has a finite radius = quantum hall droplet <ul><li>We have a solution representing a Hall droplet </li></ul>B Particles are localized at fixed radius and completely delocalized around the circle
  33. 33. Matrix description <ul><li>Vortex solutions don’t need any external source </li></ul>B We don’t add any  source The vortex pushes the boundary away a finite distance
  34. 34. Matrix description <ul><li>New solutions are obtained representing edge states </li></ul>B This is a new solution, not unitarly equivalent to the previous Diagonal radius states are mapped into non-diagonal ones
  35. 35. Matrix description We have the same canonical commutators for x a <ul><li>The quantum theory is constructed as before </li></ul>B The filling fraction is topologically quantized exactly as before The constraint relates statistics of physical states with  And additional canonical commutators for 
  36. 36. Matrix description <ul><li>The states can be found explicitly in terms of a creation and annihilation basis </li></ul>B This space of solutions is isomorphic to Laughlin wave functions!
  37. 37. Matrix description <ul><li>Some additional information </li></ul>B <ul><li>The theory is equivalent to a matrix U(1) Chern-Simons theory. </li></ul><ul><li>It can also be mapped into the Calogero or Calogero-Suterland models (Polychronakos) </li></ul><ul><li>Its quantum states have been shown to be exactly given by the Laughlin wave functions (Hellerman-Van Raamsdonk, Cappelli-Riccardi) </li></ul><ul><li>It can be extended to include multilayer system and spin of the fundamental constituent (Polychronakos-Morariu) </li></ul><ul><li>It is not known how to use this model to describe more general filling fractions p/q </li></ul>
  38. 38. Matrix description <ul><li>This theory can describe finite samples. </li></ul><ul><li>It has vortex (quasihole and quasiparticle) solutions included in the theory without adding any external source. </li></ul><ul><li>The charge of the vortex is quantized in units of the filling fraction. </li></ul><ul><li>The vortices have fractional statistics. </li></ul><ul><li>The particles have statistics according to the filling fraction. </li></ul><ul><li>The filling fraction is quantized as 1/n </li></ul><ul><li>We have perturbations describing edge states </li></ul><ul><li>The states of the theory are Laughlin wave functions! </li></ul><ul><li>It does not apply to more general filling fractions </li></ul><ul><li>Successes and limitations </li></ul>
  39. 39. Conclusions and outlook <ul><li>The Chern-Simons Matrix Model solves many of the problems of the previous formulations of FQHE. </li></ul><ul><li>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 </li></ul><ul><li>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 </li></ul><ul><li>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. </li></ul><ul><li>What’s left? </li></ul>
  40. 40. Thanks!

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