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arXiv:cond-mat/9910490 · physica status solidi (b)

How Localized is an Extended Quantum System?

Gerardo Ortíz, Armando A. Aligia

Published Jul 1, 2000• 0 citations

Abstract

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.

DOI: 10.1002/1521-3951(200007)220:1<737::aid-pssb737>3.3.co;2-0

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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.

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.

We propose a generalized quantum geometric tenor to understand topological quantum phase transitions, which can be defined on the parameter space with the adiabatic evolution of a quantum many-body system. The generalized quantum geometric tenor contains two different local measurements, the non-Abelian Riemannian metric and the non-Abelian Berry curvature, which are recognized as two natural geometric characterizations for the change in the ground-state properties when the parameter of the Hamiltonian varies. Our results show the symmetry-breaking and topological quantum phase transitions can be understood as the singular behavior of the local and topological properties of the quantum geometric tenor in the thermodynamic limit.

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $\ensuremath{\tau}$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $\ensuremath{\tau}$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if, by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $\ensuremath{\tau}$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

Many body localization (MBL) represents a unique physical phenomenon, providing a testing ground for exploring thermalization, or more precisely its failure. Here we characterize the MBL regime geometrically by the many-body quantum metric (MBQM), defined in the parameter space of twist boundary, and the localization parameter as defined in the modern theory of polarization and insulators. First, we demonstrate that the quantum metric can be used to characterize disordered insulating states by applying this theoretical framework to excited states of the 1D Anderson insulator. There we observe that the MBQM and localization parameter are related in finite realizations despite the states being gapless in the thermodynamic limit. Then, we consider a disordered 1D Bose-Hubbard model and find that we can characterize the ergodic-MBL crossover by comparing the MBQM and localization parameter. We find that we can extract a natural localization length in the MBL regime that characterizes the real space spread of the wave function and can be measured by extracting the quantum metric. Our analysis provides complementary insight into the MBL regime focusing on its insulating properties and providing a localization length whose definition is consistent across a range of insulating phases.