Abstract
The opening of a charge gap driven by interaction is a fingerprint of the transition to a Mott insulating phase. In strongly correlated low-dimensional quantum systems, it can be associated to the ordering of hidden non-local operators. For Fermionic 1D models, in the presence of spin–charge separation and short-ranged interaction, a bosonization analysis proves that such operators are the parity and/or string charge operators. In fact, a finite fractional non-local parity charge order is also capable of characterizing some two-dimensional Mott insulators, in both the Fermionic and the bosonic cases. When string charge order takes place in 1D, degenerate edge modes with fractional charge appear, peculiar of a topological insulator. In this article, we review the above framework, and we test it to investigate through density-matrix-renormalization-group (DMRG) numerical analysis the robustness of both hidden orders at half-filling in the 1D Fermionic Hubbard model extended with long range density-density interaction. The preliminary results obtained at finite size including several neighbors in the case of dipolar, screened and unscreened repulsive Coulomb interactions, confirm the phase diagram of the standard extended Hubbard model. Besides the trivial Mott phase, the bond ordered and charge density wave insulating phases are also not destroyed by longer ranged interaction, and still manifest hidden non-local orders.
Highlights
As predicted by sir Neville Mott almost seventy years ago [1], interaction is capable of opening energy gaps in bands of otherwise conducting materials, so that at sufficiently low temperature they become insulating
For Fermions in 1D, it was proved on more general grounds by means of a bosonization analysis and a renormalization-group (RG) study of the continuum limit that, in the case of spin–charge separation and local interaction, in each channel, the two possible gapped phases are univocally associated to the non-vanishing of the expectation value of one of two hidden NL operators: string or parity [2,5,6]
We review some results on the capability of hidden non-local orders, described by string and parity operators, of giving an exhaustive characterization of the insulating states induced by short range interaction in low-dimensional Fermionic materials
Summary
As predicted by sir Neville Mott almost seventy years ago [1], interaction is capable of opening energy gaps in bands of otherwise conducting materials, so that at sufficiently low temperature they become insulating. For Fermions in 1D, it was proved on more general grounds by means of a bosonization analysis and a renormalization-group (RG) study of the continuum limit that, in the case of spin–charge separation and local interaction, in each (spin or charge) channel, the two possible gapped phases are univocally associated to the non-vanishing of the expectation value of one of two hidden NL operators: string or parity [2,5,6]. For more general applications in both condensed matter and atomic physics, it would be interesting to understand how such orders and phases are modified by the presence of a long-ranged interaction, i.e., an interaction decaying with distance r as r −α , with α > 0 This is the case for instance for both dipolar interaction (α = 3), and unscreened Coulomb repulsion (α = 1). We explore at fixed large L the changes at the phase transitions among Mott, bond-ordered-wave (BOW), and charge density wave (CDW) phases, as well as the capability of NL order parameters to capture all of them
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