Quantum Hall Fluids By Andrew York 12/5/2008

Purpose and Outline Purpose: To explain field theory behind quantum Hall fluids, which leads to a description of their properties Outline Integer Hall effect Fractional Hall effect Guiding assumptions Developing the Lagrangian Interpreting the Lagrangian Understanding quasiparticles 12/5/2008

A Quantum Hall System B Collection of electrons bound to a plane, under influence of magnetic field B Magnetic field normal to plane, strong enough to align all electrons (treat as spinless) Simple descriptions of motion: Classical: QM: after setting r Electrons move in Larmor circles described by Clearly, something interesting when 12/5/2008

Integer Hall Effect Textbook case of a spinless electron in a magnetic field Wave equation: Energy levels: Degeneracy: Define “filling factor:” Clearly for integer values of , system will be incompressible Attempts to compress will reduce area and degeneracy, forcing electrons into higher energy level (keep in mind energy spacing of ) This is known as “integer Hall effect” 12/5/2008

Fractional Hall Effect Surprising experimental discovery that Hall fluids prove incompressible for filling factors with odd-denominator fractions Better description of system starts from following premises: Still confined to plane (2+1 dimensions) E-M current conserved, Descriptive field theory will follow from good local Lagrangian Important physics occurs at large distances and large time (lowest dimension terms will dominate the Lagrangian) Parity and time reversal are broken by the external magnetic field 12/5/2008

Describing Current Premises 1)* and 2)** are enough to conclude This is because Divergence = 0 implies current is the curl of something else Note gauge invariance, constant under normalization * - 1) requires 2+1 dimensions ** - 2) requires 12/5/2008

Guessing the Lagrangian 3)* and 4)** motivate us to guess the local Lagrangian with lowest order gauge invariant terms Self-interaction term: Coupling to external potential A: Quasiparticle*** interaction: Lagrangian: * - 3) states field theory follows from good local Lagrangian ** - 4) implies only lowest order terms matter in Lagrangian *** - “quasiparticle” effects are effects due to many body interaction 12/5/2008

Simplifying the Lagrangian Reorder coupling term (after integrating by parts and dropping the surface term) Defining we get Integrate using identity Re-expand using 12/5/2008

Interpreting the Lagrangian Field coupling Quasiparticle coupling Quasiparticle coupling to field From field coupling term, we deduce implies excess density of electrons related to variation in magnetic field by implies electric field produces current in the orthogonal direction, so From quasiparticle coupling to field, we deduce charge of quasiparticle is 1/k 12/5/2008

Interpreting the Quasiparticle Term Quasiparticle effects are effects due to many-body interaction (structure in the system ) In this case, quasiparticles are groups of electrons or electron holes Partially filled states can in principle be filled any way, but in practice will fall into evenly-distributed ground state due to electron repulsion System ground state can be described by distribution of quasiparticles Comparing quasiparticle term to Hopf term Statistics parameter describes the exchange of individual quasiparticles Phase factor describes the exchange of groups of k quasiparticles, and must be an odd integer for fermions Thus, k must be an odd integer, and from earlier indicates system must have odd-denominator filling 12/5/2008