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Quantum Hall Fluids Andrew York 11/10/08 Phys 611, Siopsis The study of topological quantum fluids has emerged over the last decade or so as an interesting application of Quantum Field Theory. The Hall fluid is an example of one such fluid, and describes a system in which a collection of electrons move in a plane in the presence of a magnetic field B directed normal to the plane. The magnetic field is strong enough to align all the electron spins so they may be treated as spinless fermions. In such a system, electrons take up a finite amount of space. Classically, we would say they rotate in Larmor circles (where evB mv 2 / r ) and a simple description using quantization of angular momentum mvr eBr 2 2 (after setting 1, h 2 ) implies each electron would take up a minimum area of r 2 ~ 2 2 / eB . Further, one electron’s space cannot overlap with another’s (they are fermions). Thus, a quantum Hall fluid must have enough space to allocate to each electron in the system. This is a simple description, but even from this it is obvious that the system becomes interesting when the total available area A ~ N er 2 ~ N e (2 2 / eB) , where N e is the number of electrons in the system. This paper will examine in some detail the properties of Hall fluids and the physical justification for these properties. First I will explore the integer Hall effect, which is the already obvious special circumstance that arises when the total area available is equal to the area required by the individual electrons in the fluid. Then the bulk of the paper will be devoted to understanding the fractional Hall effect, which occurs when the surface area contains an odd fraction of the number of electrons needed to fill it completely. The textbook case of a spinless electron in a magnetic field is defined by the wave equation: (1) [( x ieAx ) 2 ( y ieAy ) 2 ] 2mE 1 eB With the energy states (solved by Landau years ago) E n (n ) , n 0,1,2... known 2 m as the nth Landau level. Because the Larmor circles may be placed anywhere, each Landau level has a degeneracy of BA / 2 (where A is the area of the system). Thus we can define a filling factor N e /( BA / 2 ) that represents the number of completely filled degenerate levels. When is an integer, energy levels are completely filled and were we to add any additional electrons they would have to go into the +1 level. Clearly then, the Hall fluid is incompressible for integer values of . Any attempt to compress it would reduce the area (A), reduce the degeneracy ( BA / 2 ), and force some of the electrons into the next energy level (costing lots of energy, since the energy eB spacing is ). This incompressibility at integer filling factors is known as the integer m quantum Hall effect. Scientists were taken by surprise when experiments yielded yet another incompressible quantum Hall fluid that occurred when the filling factor was an odd- 1 1 denominator fraction ( , , etc.). This is known as the fractional quantum Hall 3 5 effect, and requires a more sophisticated field theory of the Hall fluid. This paper will attempt to explain the fractional Hall effect as the natural consequence of a few general principles of the Hall system. This treatment of the Hall system starts with the following premises, which are either obviously true or assumed to be true: 1) The system is a 2+1 dimensional spacetime (confined to a plane). 2) E-M current is conserved ( J 0 ). 3) A descriptive field theory should follow from the development of a good local Lagrangian. 4) The important physics occurs at large distances and large time (small wave number and low frequency). In other words, the lowest dimension terms will dominate the Lagrangian. a 5) Parity and time reversal are broken by the external magnetic field. 1) and 2) above are enough to conclude that the current can be written as the curl of a vector potential: 1 J a (2) 2 This follows from the fact that whenever the divergence of something is 0 in 31 dimensions, it must be the curl of something else (the factor is for normalization). It 2 is worth noting at this point that transforming a by a a doesn’t change the current, which means a is a gauge potential. Following 3) above, we should begin guessing the Lagrangian. It must be gauge invariant, so the 2D term a a won’t work. The simplest gauge-invariant term is the 3D term a a , so the first term in the Lagrangian is: L k a a 4 (3) 1 is normalization 4 over 3 dimensions. According to principle 4) above, we can ignore the higher dimensional terms that may be in the Lagrangian. We also know this system’s current couples to an external gauge potential A , so using equation (2) above we add a term to Where k is an as-yet-undetermined dimensionless parameter and 1 A a to get (after integrating by parts and 2 dropping a surface term A a ): the Lagrangian A J L k 1 k 1 a a A a a a a A 4 2 4 2 (4) Last, it is important to note that “quasiparticles” are a fundamental concept in condensed matter physics. In this context, quasiparticle effects are the effects due to the many-body interactions occurring in the medium. This is represented by a term a j in the Lagrangian. The complete Lagrangian (given assumption 4) above) is now: k 1 L a a a j a A (5) 4 2 ~ This Lagrangian can be simplified somewhat. Defining j j (1 / 2 ) A (5) becomes: k ~ L a a a j (6) 4 Then using the identity (central to QFT) K De 1 K J 2 k ~ , J j , and integrating (6) over 2 ~ L j k 2 e 1 J K 1 J 2 and in this case a , we get: ~ j 2 Which allows us to use ~ and re-expand to: 1 1 L A A A j j j 4k k k 2 (7) (8) In this Lagrangian, there is an AA term, an Aj term, and a jj term. From the AA term, we can deduce by varying A that the electromagnetic current is: 1 J em A (9) 4k The 0 part of this current tells us that an excess density of electrons is related to a local fluctuation of the magnetic field by N e (1 / 2k )B and comparing with the old definition of filling factor N e /( BA / 2 ) it becomes clear that the filling factor is associated with 1/k, and from the i component that an electric field produces current in the orthogonal direction, so 1 / k xy . The Aj term tells us the electric charge of the quasiparticle is 1/k. At this point it is instructive to step back and try to understand exactly what these quasiparticles are. Essentially, they must be collections of particles (in this case either electrons or holes of electrons) that move around in the system and interact like particles with a charge of 1/k. Imagining the system, we have a plane partially filled with electrons orbiting in Larmor circles. These electrons can in principle be distributed any way possible so long as they are not overlapping, but in practice it makes sense that electron repulsion will cause them to settle into some sort of unique ground state in which they are uniformly distributed across the plane. Such ground states are only possible for certain filling factors (as was discovered experimentally). These quasiparticles represent the hole or electron structures necessary to describe the system ground state. Since they behave like particles and cannot overlap, they should obey the rules for fermions. Their states can be interchanged with a phase factor determined by the Hopf term common to all particles. The Hopf term looks like: 1 j (10) LHopf j 2 4 1 And comparing (10) with (7), it is clear that , and thus the statistics parameter 4 k 1 that describes the interchange of particle states is related by 1 for a single 4 k quasiparticle. When exchanging k quasiparticles with k quasiparticles, we would expect 1 to pick up a phase k 1 k 2 k 2 k . k must be an odd integer for fermions, so k k must be an odd integer as well, and thus 1/ k will be an odd-denominator fraction. This explains the fractional quantum Hall effect.