We develop the theory and practical expressions for the full quantum-mechanical distribution of the intrinsic macroscopic polarization of an insulator in terms of the ground state wavefunction. The central quantity is a cumulant generating function which yields, upon successive differentiation, all the cumulants and moments of the probability distribution of the center of mass X/N of the electrons, defined appropriately to remain valid for extended systems obeying twisted boundary conditions. The first moment is the average polarization, where we recover the well-known Berry phase expression. The second cumulant gives the mean-square fluctuation of the polarization, which defines an electronic localization length ξi along each direction i: ξ2 )/N . It follows from the fluctuation-dissipation theorem that in the thermodynamic limit ξi diverges for metals and is a finite, measurable quantity for insulators. It is possible to define for insulators maximally-localized "many-body Wannier functions", which for large N become localized in disconnected regions of the high-dimensional configuration space, establishing a direct connection with Kohn's theory of the insulating state. Interestingly, the expression for ξ 2 i , which involves the second derivative of the wavefunction with respect to the boundary conditions, is directly analogous to Kohn's formula for the "Drude weight" as the second derivative of the energy.
Quantum Monte Carlo methods are used to study a quantum phase transition in a 1D Hubbard model with a staggered ionic potential (∆). Using recently formulated methods, the electronic polarization and localization are determined directly from the correlated ground state wavefunction and compared to results of previous work using exact diagonalization and Hartree-Fock. We find that the model undergoes a thermodynamic transition from a band insulator (BI) to a broken-symmetry bond ordered (BO) phase as the ratio of U/∆ is increased. Since it is known that at ∆ = 0 the usual Hubbard model is a Mott insulator (MI) with no long-range order, we have searched for a second transition to this state by (i) increasing U at fixed ∆ and (ii) decreasing ∆ at fixed U. We find no transition from the BO to MI state, and we propose that the MI state in 1D is unstable to bond ordering under the addition of any finite ionic potential ∆. In real 1D systems the symmetric MI phase is never stable and the transition is from a symmetric BI phase to a dimerized BO phase, with a metallic point at the transition.
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