# Electronic and Magnetic Properties of Metallic Phases under Coexisting Short-Range Interaction and Diagonal Disorder

Electronic and Magnetic Properties of Metallic Phases under Coexisting Short-Range Interaction and Diagonal Disorder

Hiroshi SHINAOKA^{†}^{†}thanks: E-mail address: h.shinaoka@aist.go.jp and Masatoshi IMADA

Nanosystem Research Institute, AIST, Tsukuba 305-8568

CREST, JST, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656

Department of Applied Physics, The University of Tokyo, Tokyo 113-8656

(Received July 12, 2019)

We study a three-dimensional Anderson-Hubbard model under the coexistence of short-range interaction and diagonal disorder within the Hartree-Fock approximation. We show that the density of states at the Fermi energy is suppressed in the metallic phases near the metal-insulator transition as a proximity effect of the soft Hubbard gap in the insulating phases. The transition to the insulator is characterized by a vanishing density of states (DOS) in contrast to the formation of a quasiparticle peak at the Fermi energy obtained using the dynamical mean field theory in pure systems. Furthermore, we show that there exist frozen spin moments in the paramagnetic metal.

KEYWORDS: soft Hubbard gap, electron correlation, disorder, Anderson-Hubbard model, single-particle density of states, pseudogap, Mott transition, Anderson localization, randomness, spin glass

## 1 Introduction

Understanding the nature of metal-insulator transitions (MIT) has been a central issue in condensed matter physics for a long time [Imada98]. In particular, recently, the interplay of electron correlation and randomness has been attracting much attention experimentally and theoretically because of their inevitable coexistence in real materials.

The single-particle DOS is a typical physical quantity that characterizes not only the nature of the MITs but also the electronic and magnetic properties in their vicinity. When the electron correlation causes the MIT as Mott transition [Mott49], it opens a gap in the single-particle DOS. On the other hand, the Anderson insulator, which is driven by randomness, exhibits no gap in the single-particle DOS [Anderson58]. Such contrasting behavior of the DOS for these two types of MITs raise naturally a simple question: How does the DOS behave near Mott-Anderson transitions with coexisting electron correlation and randomness? Despite extensive theoretical studies for several decades [criticality], the nature of the Mott-Anderson transition has not yet been fully clarified.

The Anderson-Hubbard model with coexisting on-site repulsion and diagonal disorder is one of the simplest models suitable for investigating the nature of the Mott-Anderson transition. Recently, we have determined the ground-state phase diagram of the three-dimensional Anderson-Hubbard model within the Hartree-Fock (HF) approximation. Furthermore, we have found an unconventional soft gap (soft Hubbard gap) over the entire insulating phases [Shinaoka09a, Shinaoka09b]. Because only the short-range interaction is present in the Anderson-Hubbard model, the soft Hubbard gap cannot be explained by the conventional theory that attributes the formation of the soft gap to the long-range nature of the Coulomb interaction [Efros75]. Indeed, we have proposed a multivalley energy landscape, which may be characteristic of random systems, as the origin of the soft Hubbard gap. This observation of the soft gap is in clear contrast to the results of a numerical study within the dynamical mean-field theory (DMFT) [Dobrosavljevic97] and some mean-field studies [Dobrosavljevic03, Byczuk05] that indicate the absence of the soft gaps. This contradiction may be due to the insufficient treatment of spatial correlation, which is essential in the formation of the soft gap, in those studies.

On the other hand, the behavior of the DOS in metals near MITs has been one of the central issues. In particular, pseudogap phenomena observed in underdoped cuprate high- superconductors have inspired fundamental discussions on the nature of the electronic states near the Mott insulator [Timusk99]. While the pseudogap and the soft gap both cause the reduction of the DOS at the Fermi level, the mechanism and origin of the pseudogap of the cuprates have not been established, and the role of randomness in stabilizing the superconductivity remains controversial. Recent numerical studies within the cellular DMFT indicates the stabilization of a pseudogap [Zhang07] or Fermi arc [Stanescu06a, Stanescu06b, Sakai09] near the MITs in pure systems, in contrast to the single-site DMFT results [Georges96]. Because the soft gap mechanism may deepen and constructively stabilize the pseudogap formation in real experimental circumstances with the inevitable coexistence of randomness and electron correlation, even on the HF level, further studies of the metallic phases near the Mott-Anderson transition will shed new light on the interplay of electron correlation and randomness and provide insight into the pseudogap phenomena.

Spin polarization (or the formation of frozen spin moments) is another essential element in determining magnetic properties in the vicinity of the MITs, such as uniform magnetic susceptibility. Although a previous HF study on the three-dimensional Anderson-Hubbard model claimed the formation of frozen spin moments even in the paramagnetic metal [Tusch93], the analyses are limited to finite system sizes and the bulk limit was not analyzed after the extrapolation.

In this paper, we show further numerical analyses of single-particle excitations and spin polarization on the metallic side within the HF approximation.

## 2 Model and Method

The Anderson-Hubbard Hamiltonian is defined as

(\theequation) |

on lattices with sites and electrons, where is the hopping integral, the on-site repulsion, () the creation (annihilation) operator for an electron with spin on site , , and is the chemical potential. The random potential is spatially uncorrelated and assumed to follow the Gaussian type distribution with the average : (). We focus on half filling () on the cubic lattice throughout this paper. We take the lattice spacing as the length unit.

We employ the Hartree-Fock (HF) approximation, where the wave function is approximated by a single Slater determinant consisting of a set of orthonormal single-particle orbitals ( is an orbital index). The HF equation reads

(\theequation) |

where is the one-body part of the Hamiltonian and we neglect and . To find a site-dependent mean-field solution for the HF equations, we employ the iterative scheme. One typically needs from several to several tens of initial guesses to obtain convergent physical quantities such as antiferromagnetic (AF) order parameters and DOS. Here, we employ pseudo-one-dimensional unit cells of , where .

## 3 Results: Electronic and Magnetic Properties of Paramagnetic Metal

### 3.1 Density of states

In Fig. LABEL:fig:1, we present the calculated ground-state phase diagram within the HF approximation [Shinaoka09a, Shinaoka09b]. Hereafter, we take the hopping integral as the energy unit. At , the Anderson-Hubbard model undergoes a metal-insulator transition (Anderson transition) from the paramagnetic metal (PM) to the paramagnetic insulator (PI) at a finite strength of disorder, [Slevin99]. On the other hand, at , since the system is half-filled with perfect nesting, the ground state is the antiferromagnetic insulator (AFI) for any nonzero value of . Here, we discuss the ground-state phase diagram for . First, we focus on the spin degree of freedom. For , the ground state is paramagnetic near . With increasing interaction, the ground states undergo an antiferromagnetic transition at a critical point . Within the resolution of our calculation, monotonically increases as disorder strength increases. Next, we focus on the charge degrees of freedom. The ground state is insulating for , which contains AFI as well as PI (PI is usually identified as an Anderson insulator). Metallic phases identified by the divergent localization length are restricted to a dome-like region ( and ).