Thank Nanoscale for bringing to my attention a long standing puzzle in mesocopic scale condensed matter physics, the “0.7 anomaly”. The problem is the behavior of the conductance (inverse of resistance) for a quantum point contact (QPC). Two experimental papers showing the effect are 0706.0792 and cond-mat/0005082, which see. Shot noise decreases at the 0.7 anomaly just as it does at the integer conductance points, see cond-mat/0311435. A recent perturbation analysis of the situation is given by 0707.1989. This blog post gives a non perturbational calculation.
Such a point contact is a region which is so small that it can hold at most a single electron. One controls the size of the conductance region by applying a voltage to a gate. One expects that the conductance will be a multiple of G0 = 1/13K ohms; this works for high conductance values (increasing to saturation) but an anomaly appears near cutoff.
This really is basic physics and should be well understood. The computer that I (and you, assuming you’re reading this in 2008) are using are built from CMOS logic gates. The QPC effect occurs when such a gate is operated near its cutoff point. There are billions of billions of these gates currently in operation on this planet (operating in cutoff and saturation); by all expectations they should have been well understood many decades ago.
However, it turns out that the situation is difficult to analyze with the usual tools of quantum mechanics. One ends up with a highly nonlinear situation involving “quasibound” states. Since the density matrix formalism I work with is specifically designed to solve highly nonlinear bound state QFT problems without recourse to the usual perturbation theory, this is a natural place to explore its application. In addition, this could be a good first application of MUB (mutually unbiased bases) to quantum theory.
The 0.7 anomaly is well described in the introductory APS article A Mesoscopic Mystery:
Researchers had thought that a quantum point contact (QPC) was essentially a “particle in a box,” whose properties are the stuff of sophomore physics classes. The conductance between the two electron reservoirs increases in a stepwise fashion as the width of the connecting channel is increased, paralleling the ground and excited states of an atom. “Everyone thought, ‘My gosh, what could be simpler than that?'” says Ned Wingreen of the NEC Laboratories America, Inc., in Princeton. Then in 1996 researchers saw an additional step where it shouldn’t be if the system were truly operating by simple rules. The so-called 0.7 anomaly, named for its conductance value in units of fundamental constants, means “what we thought is best understood in mesoscopic physics is least understood,” says Wingreen.
Most attempts to crack the problem have revolved around electron spin, which comes in two values, up and down. When a magnetic field is applied to the QPC, the conductance levels “split”–new steps appear between the old ones–because the magnetic field affects electrons differently depending on their spin. The 0.7 step remains even at zero magnetic field, so researchers have proposed that interactions among the electrons could be producing an intrinsic magnetic field that spontaneously aligns spins and causes some splitting.
To conduct electricity, an electron begins on one side of the QPC and eventually ends up on the other side. This fits naturally into density matrix theory as an example of scattering. The electron’s initial state is on one side of the QPC and its final state is on the other. If we make the assumption that the spin of the electron is unchanged by its interaction with the QPC, we end up with two processes described: the propagation of spin up electrons, and the propagation of spin down electrons. The density (scattering) matrices for these processes, written with the usual Pauli spin matrix representations, are as follows:
The point of density matrix theory is that one should write quantum states as processes, with input states and output states. This is in comparison to the usual state vector theory where one thinks of quantum states as existing at a particular time, and therefore as suitable to be represented by just the “input state” or “output state”. The theoretical reason for preferring the density matrix point of view is that when one attempts to understand quantum processes in terms of individual states, the individual states become nonsensical (think of the “which way” two-slit experiments or the EPR paradox). The practical reason for preferring the density matrix form is that it makes calculations of interacting systems much simpler.
In the case of the conductance of a QPC, it is clear that at high enough conductance levels, the usual scheme works beautifully and the conductance is quantized in multiples of Q0. The usual way one analyzes N-particle electron states in quantum mechanics is to first choose a spin direction, typically the z-direction, and one defines the quantum states in that direction, i.e. spin up and spin down. Next one takes N copies of this and works with the tensor product. One keeps only those combinations that are compatible with the Pauli exclusion principle. Because of the way the Pauli exclusion principle is written, one must assume that all the N copies are taken with respect to the same spin direction. And this theory works fine for the conductance of the QPC. Where the standard theory breaks down is in the conductance that exists below the level for a single electron.
Spin up / down becomes spin anything
When one has less than a single electron, the conductance bands need not have the simple spin up / spin down structure assumed for the multi-particle condition. Since the QPC interacts with the electron, we can have far more complicated scattering conditions than simply assuming that spin up scatters into spin up, and spin down scatters into spin down.
For example, one could have an electron which arrives spin polarized in the x direction, and leaves spin polarized in the y direction. We will assume that all these sorts of processes contribute to conductance. As with any QFT calculation, we will sum over all possibilities and then compute an amplitude, take its magnitude, and square it. But we will do this in the context of density matrices. For density matrices, the “take magnitude and square it” is eliminated in how the density matrix is put together, and instead becomes a “take trace.”
Mutually Unbiased Bases
To allow general spin changes in the scattering through the QPC, we will let the incoming electron be spin +x, +y, or +z, and the outgoing electron also have any of these spins. This means that we will be discussing 9 scattering processes. We can arrange these 9 processes into a square matrix:
In choosing +x, +y, and +z, and not using -x, -y, or -z, we have avoided including two quantum states that are incompatible such as both spin up and spin down. This limits us to 3 conditions on the incoming electron and 3 on the outgoing electron. Just as with the spin up and spin down channels in the usual QM conductance calculation, the choices made above, to take the + signs, is arbitrary; but the calculation does not depend on it (by symmetry).
The next step in the calculation is to require that the density matrix be self consistent. In the usual (pure state) density matrices, this amounts to the idempotency requirement: BB = B. For the situation where the conductance is handled through the nine MUB scattering conditions given in matrix form, the equivalent requirement is that this matrix square to itself. (Note that matrix multiplication sends +x final states into +x initial states, etc., as is required under the assumption that the matrix is the only source of spin changes to the electrons. And note that matrix multiplication assumption works only in momentum space. This is due to how convolutions of Feynman diagrams becomes multiples of the Fourier transforms.)
In the case of QPC conductance, matrix multiplication is unphysical in that any single electron only goes through the scattering process once, however, conductance goes both ways so what we are really doing here is requiring that the scattering matrix be compatible for an electron that moves back and forth through the QPC.
In the situation where there is no applied magnetic field, the +x, +y, and +z directions are equivalent and so we expect the matrix solution to be symmetric under permutions of these directions. This amounts to requiring that the matrix be circulant. What are the 3×3 matrices of complex numbers that are primitive idempotent and circulant? This is an algebra problem.
Those who have been following my work know that the solution to the 3×3 circulant primitive idempotent problem is how I found the generalization of Koide’s mass formula for the neutrinos. The algebraic solution to the problem is given in Chapter 3 of my incomplete book on density matrix theory, which is still available from Lulu if you want a hard copy. One finds that there are three primitive idempotent solutions:
where w is any of the three cubed root of unity, .
In the case of the Koide formula, the three solutions correspond to the 3 generations of fermions. For our purposes, the three solutions are simply three different ways of getting electrons through a QPC. In the method of Feynman diagrams, one takes sums over the complex numbers that contribute to an amplitude, and then computes the square of the magnitude of that complex sum. This works because the contributions to the usual Feynman diagram are all compatible diagrams. That is, they begin and end with the same states.
For the case of a scattering matrix of quantum processes, we cannot replace the entries with complex numbers and expect to get the correct result. Instead, we need to correct for “Berry phase” and for the probabilities that arise in spinor probability transitions. For example, the process z->x followed by x->z is not z->z but instead, since , we get 0.5 z->z. In MUB theory, the factor of 1/2 comes from the transition probabilities between two bases of a MUB. And more complicated series, such as z->x->y->z in addition to the spinor probability factors, end up with a complex phase.
To account for transition probabilties and Berry phase, we have to modify the three solutions to the circulant 3×3 matrix problem as follows:
where . The square roots of two are there because when you multiply two off diagonal entries to give another entry, there will be a loss of amplitude due to transition probabilities. The angle is there to fix the Berry phases. This is 1/3 the angle you get when you look at x->y->z relative to x->z. The book on density matrix theory goes into more detail on these calculations.
I suppose that I can write down the solution in a form that the reader can use to verify that the above alterations do produce an idempotent collection of scattering matrix entries. Let be the multiple of the (x,y) entry of the 3×3 matrix and similar for the other 8 entries. Then is a complex multiple of where all calculations are to be done with “x,y,z” as the 2×2 Pauli spin matrices.
The square of the 3×3 matrix requires that the following nine equations be solved:
The above is just matrix multiplication applied to the matrix of A and making the assignment A = A^2.
The claim is that the above equations are solved by
for the diagonal terms and
for the off diagonal terms, where “n” is 0, 1, or 2.
The reader can substitute the Pauli spin matrices for x, y, and z and verify that these substitutions solve the nine equations for matrix squaring. Or you can take my word for it. (But be warned, sometimes I get the sign wrong.) The reader will find that in verifying the diagonal equations the complex phases all cancel. The off diagonal terms that contribute to the diagaonal term will get their square root of 2 cancelled by the transition probability for x->z->x. When verifying the off diagonal terms, the reader will find that the square roots of two all cancel. The Berry phase, which comes from things like x->y->z being written as a complex multiple of x->z, will introduce a factor of i. This factor will be cancelled exactly by the way that the phases are distributed among the contributions.
The diagonal entries of the 3×3 matrix has entries that are the usual density matrices. Their traces are all unity, so the overall trace of the diagonal elements of the primitive idempotents is (1/3) x 3 = 1, even after correcting for probabilities and Berry phases. However, in the 3×3 matrix form, the off diagonal elements are also as important as the diagonal entries and must be added into the trace. Another way of describing this is that in geometric form, the trace is nothing but the scalar part of a transition. The off diagonal transitions also have scalar parts and so must be included.
The on diagonal density matrices are given by things like (1+x)/2 where “x” is the Pauli spin matrix. There scalar part is 1/2, and this gives a trace of 1 because these are 2×2 matrices. The off diagonal entries are products of these sorts of things and look like (1+x)/2 (1+y)/2 = (1+x+y+xy)/4. These have scalar part 1/4 and therefore have traces of 1/2. Thus the off diagonal entries contribute to the trace to the extent of half as much as the diagonal entries.
In calculating the conribution from the off diagonal terms, the exponentials will add in pairs to give twice the cosine. This twice cancels the 1/2 that comes from the smaller trace of the off diagonal terms relative to the diagonal terms. So the contribution from one row of off diagaonal terms is . Three of these eliminates the 1/3, and adding in the contribution to the trace from the diagonal (which is 1), the total trace is for n=0,1,2. These give values for the traces for the three mixed spin transfer conditions as follows:
Note that the above three traces add to 3. This is due to the fact that we are using 3×3 matrices and the trace of unity is therefore 3. Since the Pauli matrices are 2×2 matrices, the trace of their unity is 2 and the sum of the transition probabilities for spin up to spin up and spin down to spin down is 2. If we redid the calculation assuming two perpendicular spin states instead of three, we’d get two traces that added to two. One would be larger than 1, the other smaller.
If the three transitions did not interact, i.e. if x->x, y->y, and z->z were independent, non interacting processes, we would expect them all to have equal trace (and equal conductance). Allowing the three channels to interact has split these three equal conductances into 0.00, 0.63, and 2.37. The 0.00 trace is easy, its conductance is zero, it does not conduct. The 2.37 trace does not contribute because its conductance value is above the conductance of a single spin up channel, which we know has value 1. This leaves only the 0.633975 channel as the “0.7 anomaly.” It should be noted that some researchers put the 0.7 anomaly at 0.6, for example see, “the so called 0.7 (in our case
0.6) anomaly” in 0712.1672. So the 0.633975 computation is approximately the right value.
The assumptions made here is that the QPC as a scattering system for electrons, randomizes the spin orientation according to MUB principles. One can suppose that this scattering decreases as the scattering time is decreased. If the scattering were eliminated entirely, the off diagonal terms would be eliminated and the three trace values 0.00, 0.63, and 2.37 would converge all to 1.00.
There are some hints of this sort of behavior in the experimental data. See cond-mat/0005082 for an indication that the 0.7 anomaly increases to 0.9 as V_sd (also called the “bias voltage” ) is increased. V_sd is the voltage between the source and drain. This is the driving voltage that causes current flow. As it increases away from zero, the current increases, and consequently the ability of the electron to interact with itself decreases. Under this condition, with this theory, it is entirely expected that the 0.7 anomaly moves towards 1.
Another way to alter the 0.7 anomaly is to change the temperature. The modification of the spin of an electron travelling through the QPC is presumably due to thermal interactions. As the temperature decreases, we can expect that these interactions will decrease. The effect will be to reduce the 0.7 anomaly and this is also seen in the data of the paper linked in here.