Research Projects

In this paper the center of mass momentum operator is derived from its generating function. It is emphasized that this operator describes current in normal conductors. A state which is an eigenstate of this operator has a finite Drude weight and is also delocalized. An external potential is a necessary but not sufficient condition for insulation. Whether a model corresponds to an insulator or conductor depends on the ratio of the the external potential to the energy scale of the underlying interacting system. It is also shown that a normal conductor of finite size exhibits flux quantization, and it is only in the thermodynamic limit that the flux becomes continuous. The Hubbard model is analyzed, and it is demonstrated that it can not exhibit insulating behaviour. Both the large $U$ and small $U$ states are perfect conducting states, the difference is that the former does not exhibit flux quantization whereas the latter does.

An expansion, similar to the cumulant expansion in probability theory, is carried out for the Bargmann invariant, which is the quantity from which the Berry phase can be derived. The derivation shows that the first term in the expansion corresponds to the Berry phase itself, the higher order terms can be interpreted as the associated cumulants; spread, skew, kurtosis, etc. The gauge invariance of all of these quantities is also demonstrated.

This paper deals with three quantities used in the theory of transport; the Drude weight, the Meissner weight, and the non-classical part of the rotational inertia of bosonic superfluids. These three quantities have nearly identical (apart from constants) mathematical expressions: they are all proportional to the second derivative of the ground state energy with respect to a momentum shift. How can one distinguish these three quantities? The first derivative of the ground state energy with respect to the momentum shift corresponds to the current. It turns out that the current can be cast in terms of a Berry phase. When this is done, it becomes obvious that there is an ambiguity in the definition of the current: one can use the total momentum shift operator, but one can also use a single-particle momentum shift operator and sum over all particles, or use a momentum shift operator for pairs of particles and sum over these pairs. From the current expressed in the first manner, one can derive the Drude weight, the second gives the rotational inertia of a bosonic superfluid, the third the Meissner weight. One can justify this correspondence by showing that the Meissner weight obtained in this manner is finite for systems which exhibit off-diagonal long-range order (ODLRO) in the second order reduced density matrix, whereas the non-classical rotational inertia will do that already for the first order reduced density matrix. The Drude weight obtained in this manner is also sensitive to ODLRO, but only for a reduced density matrix of order N, where N is a thermodynamically large number. Apart from these results, a criterion to distinguish superfluids from Bose-Einstein condensates emerges. One diagonalizes the single particle reduced density matrix and considers the eigenstates. The former (as is already known) consists of a thermodynamically large number of particles occupying single particle eigenstates of the momentum with zero momentum. The latter consists of thermodynamically large number of particles occupying single particle momentum eigenstates with any eigenvalue.

In this paper it is shown that the Drude weight can be expressed in terms of a topological invariant. The topological invariant is very similar to the TKNN invariant which appears in the expression of the Hall conductance. The TKNN invariant results from integrating around the two-dimensional Brillouin zone. The invariant associated with the Drude weight results from an integral around a rectangle, one side of which is the total momentum, the other is the total position. The Drude weight expressed in this manner furnishes a simple proof of the connection between insulation and many-body localization.