Condensed Matter Theory Laboratory

Outline

The ultimate goal of our research is to understand theoretically various properties of materials of macroscopic scale from the physical laws that govern the microscopic world. Many-electron systems go into ordered phases with spontaneous symmetry breaking at low temperatures. Typical examples are superconductivity and magnetism found in strongly-correlated electron systems like transition-metal oxides and molecular conductors, which are our major research subjects.

In strongly-frustrated quantum spin systems such as triangular or kagome antiferromagnets, an exotic quantum state, instead of a magnetic ordered state, is expected to emerge. For example, we have recently considered spin nematic and vector chiral ordered states. Furthermore, we study states with some kind of topological order: topological insulators and superconductors.

We also study universal properties at localization-delocalization transitions of electrons moving in random potential, which is a disorder-induced quantum phase transition.

Recent Research Topic

Universality emerging from randomness

In a solid containing many impurities and lattice defects, electronic transport is hindered by frequent scattering of electrons by impurities and defects. As disorder is increased, electrons become Anderson-localized due to quantum interference, and a transition from a metal to an insulator occurs. Such a metal-insulator transition is a quantum phase transition driven by disorder. At a metal-insulator transition point, the wave function of an electron has a mixed character of extended and localized states, showing complicated, fractal-like spatial variations (Fig. 1). We want to understand universal properties of critical behaviors at this type of phase transition. We have numerically studied critical exponents of diverging localization length and multifractal spectra of critical wave functions for the symplectic class (of strong spin-orbit interactions) and the unitary class (quantum Hall systems), both of which have critical points in two spatial dimensions.

It is known that conformal symmetry emerges at a critical point of continuous phase transition in two-dimensional disorder-free systems, and its universal critical behavior is described by a conformal field theory. However, it was not known whether conformal symmetry is present at disorder-induced continuous phase transitions. We have shown numerically that multifractal spectra of critical wave functions at edges and those at corners are related through a simple formula dictated by conformal invariance (Fig. 2). Furthermore, we have shown that a multifractal exponent in two dimensions and a localization length in quasi-one dimension are related through logarithmic transformation. These results strongly suggest presence of conformal symmetry and furthermore underlying (non-unitary) conformal field theories. This is a small step forward for resolving the very difficult question: what are the (non-unitary) conformal field theories describing two-dimensional quantum phase transitions induced by disorder, such as Anderson transitions.

An effective theory of transport in disordered systems is a nonlinear sigma model. It was known from homotopy theory that the nonlinear sigma model of the two-dimensional symplectic class may have a Z2 topological term. We have shown that for Dirac fermions moving in a random scalar potential, a Z2 topological term is indeed present. These Dirac fermions appear as gapless surface states of a three-dimensional Z2 topological insulator and are never localized because of non-perturbative effects of the Z2 topological term. As a natural extension of this idea, by examining the presence or absence of a topological term for arbitrary single-particle Hamiltonians, we have constructed a general theory that allows us to classify, in terms of symmetries, topological insulators and superconductors in any spatial dimension. A topological insulator is a band insulator with gapless fermionic excitations with linear energy dispersion at the boundaries. A well-known example is an integer quantum Hall liquid, and, more recently, time-reversal invariant two- and three-dimensional topological insulators with Z2 topological numbers have attracted a lot of attention. In the context of Anderson localization, one can view those topological insulators as an insulator having topologically stable boundary excitations which can never be Anderson localized by disorder. Similarly, a topological superconductor (superfluid) has gapless fermionic excitations at the boundaries, and the superfluid B phase of helium 3 is an example of a topological superfluid. Here the boundary excitations are not ordinary fermions but rather exotic particles called Majorana fermions, which are interesting subjects for further study.

Fig. 1 Critical wave function amplitude

Fig. 2 Multifractal spectra f(α) of critical wave function amplitudes in the bulk (pink) and at edge (blue), 45°corner (cyan), 90°corner (red), 135°corner (green)

The solid curve is calculated from the edge spectrum (blue) using a theoretical formula imposed by conformal invariance. The theoretical curve shows good agreement with the results (circles) of direct numerical calculation.