The question of conductivity is revisited. Using the total momentum shift operator to construct the perturbed many-body Hamiltonian and ground state wave function the second derivative of the ground state energy with respect to the perturbing field is expressed in terms of the one and two-body momentum densities. The distinction between the adiabatic and envelope function derivatives, hence that between the Drude and superfluid weights, can be introduced in a straightforward manner. It is shown that a discontinuity in the momentum density leads to a contribution to the Drude weight, but not the superfluid weight, however a δ-function contribution in the two-body momentum density (such as in the BCS wave-funtion) contributes to both quantities. The connection between the discontinuity in the momentum density and localization is also demonstrated.
The Drude weight, the quantity which distinguishes metals from insulators, is proportional to the second derivative of the ground state energy with respect to a flux at zero flux.The same expression also appears in the definition of the Meissner weight, the quantity which indicates superconductivity, as well as in the definition of non-classical rotational inertia of bosonic superfluids.It is shown that the difference between these quantities depends on the interpretation of the average momentum term, which can be understood as the expectation value of the total momentum (Drude weight), the sum of the expectation values of single momenta (rotational inertia of a superfluid), or the sum over expectation values of momentum pairs (Meissner weight).This distinction appears naturally when the current from which the particular transport quantity is derived is cast in terms of shift operators.
In the absence of frustration, interacting bosons in their ground state in one or two dimensions exist either in the superfluid or insulating phases. Superfluidity corresponds to frictionless flow of the matter field, and in optical conductivity is revealed through a distinct δ-functional peak at zero frequency with the amplitude known as the Drude weight. This characteristic low-frequency feature is instead absent in insulating phases, defined by zero static optical conductivity. Here we demonstrate that bosonic particles in disordered one dimensional chains can also exist in a conducting, non-superfluid, phase when their hopping is of the dipolar type, often viewed as short-ranged in one dimension. This phase is characterized by finite static optical conductivity, followed by a broad anti-Drude peak at finite frequencies. Off-diagonal correlations are also unconventional: they feature an integrable algebraic decay for arbitrarily large values of disorder. These results do not fit the description of any known quantum phase, and strongly suggest the existence of an unusual conducting state of bosonic matter in the ground state.
Owing to the fact that the particle current operator in nonrelativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight Δ. The Drude weight can be calculated within linear response theory. It is given by the formula Δ=2πD, where D is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current autocorrelation function.
We calculate the Drude weight in the superfluid (SF) and supersolid (SS) phases of the hard-core boson (HCB) model on a square lattice using stochastic series expansion (SSE). We demonstrate from our numerical calculations that the normal phase of HCBs in two dimensions can be an ideal conductor with dissipationless transport. In two dimensions, when the ground state is a SF, the superfluid stiffness drops to zero with a Kosterlitz-Thouless type transition at
TKT. The Drude weight, though is equal to the stiffness below
TKT, surprisingly, stays finite even for a range of temperatures above
TKT indicating the nondissipative transport in the normal state of this system. In contrast to this, in a three-dimensional SF phase, where the superfluid stiffness goes to zero continuously via a second-order phase transition at
Tc, the Drude weight goes to zero at
Tc, as expected. We also calculated the Drude weight in a two-dimensional SS phase, where the charge density wave (CDW) order coexists with superfluidity. For the SS phase we studied, superfluidity is lost via a Kosterlitz-Thouless transition at
TKT and the transition temperature for the CDW order is larger than
TKT. In striped SS phase where the CDW order breaks the rotational symmetry of the lattice, the system behaves like an ideal conductor for a range of temperatures above
TKT along the lattice direction parallel to the stripes, while along the direction perpendicular to the stripes it behaves like an insulator for all
T>TKT. In contrast to this, in the star-SS phase, the Drude weight along both lattice directions goes to zero along with the superfluid stiffness and for
T>TKT we have the finite temperature phase of a CDW insulator.