Manifestation of quantum distance in flat band materials
Landau level spectrum of singular flat bands enables to measure the
distance between quantum states

Date:
August 5, 2020
Source:
Institute for Basic Science
Summary:
A research team found a way to measure the quantum distance of
Bloch states in solids by applying magnetic field.



FULL STORY ==========================================================================
The geometry of an object indicates its shape or the relationship of
its parts to each other. Did you know that the electrons in solids also
have geometric structures? In quantum mechanics, an electron in solids
takes the form of a wave with periodicity so that the periodic electronic state, so-called the Bloch state, can be characterized by specifying its
energy and crystal momentum which is proportional to its wave number. The relationship between the energy and the crystal momentum of electrons
is called the band structure of solids.

For electrons in solids, the Berry curvature and the quantum metric
of Bloch states take the role of the curvature and the distance of an
object in geometry.


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In fact, the geometry of quantum states is one of the central concepts underlying diverse physical phenomena, ranging from the celebrated
Aharonov- Bohm effect to the topological phases of matter developed more recently. For instance, the local Berry curvature is responsible for
the anomalous Hall transport while its integral over a two-dimensional
closed manifold gives the Chern number, an integer number describing
the quantized Hall conductivity.

However, compared to the physics of the Berry curvature, the effects of
the quantum metric on physical phenomena are less understood, especially
in solids, although there are several recent works proposing the physical observables related to the quantum metric. Especially, there has been no
clear guideline for searching materials in which the physical properties related with the quantum metric can be observed.

Prof. YANG Bohm-Jung and Dr. RHIM Jun-Won at the Center for Correlated
Electron Systems within the Institute for Basic Science (IBS) in Seoul
National University, Seoul, South Korea, and Dr. KIM Kyoo at the Korea
Atomic Energy Research Institute, Daejeon, South Korea, reported that
they found a way to measure the quantum distance of Bloch states in
solids by applying magnetic field. More specifically, the researchers
have examined the energy spectrum under magnetic field, called the
Landau level spectrum, of flat bands in the kagome and checkerboard
lattices, and observed anomalous Landau level spreading arising from
the flat band. Surprisingly, they found that the total energy spreading
of the flat band's Landau level is solely determined by the maximum
quantum distance between the Bloch states of the flat band. Namely,
the quantum distance of the Bloch states in solid can be measured by
applying magnetic field to two-dimensional materials with flat bands!
Recently, two-dimensional materials with flat bands have received a great attention as a new platform to realize intriguing electronic states. A
flat band indicates an electronic band structure in which the energy
does not change when crystal momentum is varied. Such intriguing flat
band structures appear in various two-dimensional lattices including
the kagome lattice, checkerboard lattice, etc. The theory group of the
IBS CCES research team realized that, in many of the flat band systems,
the Berry curvature of the Bloch states is zero due to the symmetry of
the lattice. If the Berry curvature is strictly zero, one can naturally
expect that the geometry of the Bloch states is solely determined by
the quantum metric. This interesting aspect motivated the IBS theory
team to seriously consider two-dimensional materials with flat bands
as a promising playground to study physical properties related to the
quantum metric.

In fact, the semiclassical quantization rule predicts that an ordinary parabolic band under magnetic field forms equally-spaced discrete Landau levels, and the energy difference between neighboring Landau levels
is inversely proportional to the effective mass of the electrons. When
applied to a flat band with an infinite effective mass, the semiclassical theory predicts zero Landau level spacing so that a flat band remains
inert under magnetic field. In this study, the researchers observed
a quite peculiar nature of the Landau level spectrum that is in sharp
contrast to the conventional norm. They reported that the Landau levels
of the flat bands spread into the empty region of the energy space where
no electronic states are available in the absence of magnetic field.

The researchers found that the key to such an unusual Landau level
spectrum is the fact that the flat band in the kagome and checkerboard
lattices is crossing with another parabolic band at a point. The
singularity in the wave function of the flat band arising from the band crossing point induces nontrivial geometric effect related to the quantum distance of the wave function, which in turn induces anomalous Landau
level spectrum. The first author, Dr. RHIM Jun-Won states, "Understanding
the role of the band crossing in flat bands was the key to describe
the anomalous Landau levels. This finding provides a practical way to unambiguously extract the quantum distance in solids." This study shows
that the quantum distance or quantum metric can also play crucial roles
in determining material properties as the Berry curvature does.

Contrary to the previous works, this study clearly identified
the candidate lattice systems in which the quantum metric effect is
maximized while the Berry curvature effect is minimized, and discovered
a way to directly extract the quantum distance in solids for the first
time. Considering the tremendous impact of the concept of the Berry
curvature on the understanding the properties of solids, it is natural to expect that this study may facilitate the future study about the geometric properties of solids related to the quantum metric and the search of
materials in which the related physical responses can be observed.

Prof. YANG Bohm-Jung explains that "This result would provide a critical
step towards the complete understanding of geometrical properties of
quantum states in solids. As there are many two-dimensional lattice
structures hosting flat bands, our study may trigger future research
activities for discovering novel geometrical phenomena related to the
quantum metric in various condensed matter materials."

========================================================================== Story Source: Materials provided by Institute_for_Basic_Science. Note:
Content may be edited for style and length.


========================================================================== Journal Reference:
1. Jun-Won Rhim, Kyoo Kim, Bohm-Jung Yang. Quantum distance and
anomalous
Landau levels of flat bands. Nature, 2020; 584 (7819): 59 DOI:
10.1038/ s41586-020-2540-1 ==========================================================================

Link to news story: https://www.sciencedaily.com/releases/2020/08/200805124042.htm

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