Superfluid weight and polarization amplitude in the one-dimensional bosonic Hubbard model

We calculate the superfluid weight and the polarization amplitude for the one-dimensional bosonic Hubbard model focusing on the strong-coupling regime. Other than analytic calculations we apply two methods: variational Monte Carlo based on the Baeriswyl wave function and exact diagonalization. The former gives zero superfluid response at integer filling, while the latter gives a superfluid response at finite hopping. From the polarization amplitude we derive the variance and the associated size scaling exponent. Again, the variational study does not produce a finite superfluid weight at integer filling (size scaling exponent is near one), but the Fourier transform of the polarization amplitude behaves in a similar way to the result of exact diagonalization, with a peak at small hopping, and suddenly decreasing at the insulator-superfluid transition. On the other hand, exact diagonalization studies result in a finite spread of the total position which increases with the size of the system. In the superfluid phase the size scaling exponent is two as expected. Importantly, our work addresses the ambiguities that arise in the calculation of the superfluid weight in variational calculations, and we comment on the prediction of Anderson about the superfluid response of the model at integer filling.