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arXiv:cond-mat/9810348 · Physical Review Letters

Quantum Mechanical Position Operator and Localization in Extended Systems

Armando A. Aligia, Gerardo Ortíz

Published Mar 22, 1999• 101 citations
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Abstract

We introduce a fundamental complex quantity, ${z}_{L}$, which allows us to discriminate between conducting and nonconducting thermodynamic phases in extended quantum systems. Its phase can be related to the expectation value of the position operator, while its modulus provides an appropriate definition of a localization length. The expressions are valid for any fractional particle filling. As an illustration we use ${z}_{L}$ to characterize insulator to ``superconducting'' and Mott transitions in one-dimensional lattice models with infinite on-site Coulomb repulsion at quarter filling.

DOI
10.1103/physrevlett.82.2560

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We introduce a fundamental complex quantity, zL{z}_{L}zL​, which allows us to discriminate between conducting and nonconducting thermodynamic phases in extended quantum systems. Its phase can be related to the expectation value of the position operator, while its modulus provides an appropriate definition of a localization length. The expressions are valid for any fractional particle filling. As an illustration we use zL{z}_{L}zL​ to characterize insulator to ``superconducting'' and Mott transitions in one-dimensional lattice models with infinite on-site Coulomb repulsion at quarter filling.
We elaborate on a geometric characterization of the electromagnetic properties of matter. A fundamental complex quantity, zL, is introduced to study the localization properties of extended quantum systems. zL, which allows us to discriminate between conducting and non-conducting thermodynamic phases, has an illuminating physical (and geometric) interpretation. Its phase can be related to the expectation value of the position operator (and a Berry phase), while its modulus is associated with quantum electric polarization fluctuations (and a quantum metric). We also study the scaling behavior of zL in the one-dimensional repulsive Hubbard model.
We elaborate on a geometric characterization of the electromagnetic properties of matter. A fundamental complex quantity, zL, is introduced to study the localization properties of extended quantum systems. zL, which allows us to discriminate between conducting and non-conducting thermodynamic phases, has an illuminating physical (and geometric) interpretation. Its phase can be related to the expectation value of the position operator (and a Berry phase), while its modulus is associated with quantum electric polarization fluctuations (and a quantum metric). We also study the scaling behavior of zL in the one-dimensional repulsive Hubbard model.
Insulating systems are characterized by their insensitivity to twisted boundary conditions as quantified by the charge stiffness and charge localization length. The latter quantity was shown to be related to the expectation value of the many-body position operator and serves as a universal criterion to distinguish between metals and insulators. In this work we extend these concepts to a new class of quantum systems having conserved charge and dipole moments. We refine the concept of a charge insulator by introducing notions of multipolar insulators, e.g., a charge insulator could be a dipole insulator or dipole metal. We develop a universal criterion to distinguish between these phases by extending the concept of charge stiffness and localization to analogous versions for multipole moments but with our focus on dipoles. We are able to relate the dipole localization scale to the expectation value of a recently introduced many-body quadrupole operator. This refined structure allows for the identification of phase transitions where charge remains localized but, e.g., dipoles delocalize. We illustrate the proposed criterion using several exactly solvable models that exemplify these concepts and discuss a possible realization in cold-atom systems.
Inspired by the duality picture between superconductivity (SC) and insulator in two spatial dimension (2D), we conjecture that the order parameter, suitable for characterizing 2D fermionic insulating state, is the disorder operator, usually known in the context of statistical transformation. Namely, the change of the phase of the disorder operator along a closed loop measures the particle density accommodating inside this loop. Thus, identifying this (doped) particle density with the dual counterpart of the magnetic induction in 2D SC, we can naturally introduce the disorder operator as the dual order parameter of 2D insulators. The disorder operator has a branch cut emitting from this ``vortex'' to the single infinitely far point. To test this conjecture against arbitrary two-dimensional lattice models, we have chosen this branch cut to be compatible with the periodic boundary condition and obtain a general form of its expectation value for noninteracting metal and/or insulator wave function, including gapped mean-field order wave function. Based on this expression, we observed analytically that it indeed vanishes for a wide class of band metals in the thermodynamic limit. On the other hand, it takes a finite value in insulating states, which is quantified by the localization length or the real-valued gauge invariant two form dubbed as the quantum metric tensor. When successively applied along a closed loop, our disorder operator plays the role of twisting the boundary condition of a periodic system. We argue this point by highlighting the Aharonov-Bohm phase associated with this nonlocal operator.
Coulomb repulsion between electrons moving on a frustrated lattice can give rise, at simple commensurate electronic densities, to exotic insulating phases of matter. Such a phenomenon is illustrated using an extended $t\text{\ensuremath{-}}J$ model on a planar pyrochlore lattice for which the work on the quarter-filled case [D. Poilblanc et al., Phys. Rev. B 75, 220503 (2007)] is complemented and extended to 1∕81∕81∕8 and 3∕83∕83∕8 fillings. The location of the metal-insulator transition as a function of the Coulomb repulsion is shown to depend strongly on the sign of the hopping. Quite generally, the metal-insulator transition is characterized by lattice symmetry breaking but the nature of the insulating Mott state is more complex than a simple charge density wave. Indeed, in the limit of large Coulomb repulsion, the physics can be described in the framework of (extended) quantum fully packed loop or dimer models carrying extra spin degrees of freedom. Various diagonal and off-diagonal plaquette correlation functions are computed and the low-energy spectra are analyzed in detail in order to characterize the nature of the insulating phases. We provide evidence that, as for an electronic density of n=1∕2n=1∕2n=1∕2 (quarter filling), the system at n=1∕4n=1∕4n=1∕4 or n=3∕4n=3∕4n=3∕4 exhibits also plaquette order by forming a (lattice rotationally invariant) resonant singlet pair crystal, although with a quadrupling of the lattice unit cell (instead of a doubling for n=1∕2n=1∕2n=1∕2) and a fourfold degenerate ground state. Interestingly, qualitative differences with the bosonic analog (e.g., known to exhibit columnar order at n=1∕4n=1∕4n=1∕4) emphasize the important role of the spin degrees of freedom in, e.g., stabilizing plaquette phases with respect to rotational symmetry-breaking phases.