# Ground-state atlas of a three-dimensional semimetal in the quantum limit

## TL;DR

This paper provides a comprehensive theoretical phase diagram of a three-dimensional semimetal in the quantum limit, revealing various ground states influenced by electron interactions and screening effects, with implications for experimental observations.

## Contribution

It introduces an unbiased theoretical framework combining parquet RG and mean-field methods to map ground states of 3D semimetals under high magnetic fields, considering both repulsive and attractive interactions.

## Key findings

- Ground state varies from excitonic insulator to spin density wave depending on screening.
- Topological excitonic insulator supports massless Dirac fermions on the surface.
- Surface SdH oscillations exhibit a $
oot H_{\perp}$ dependence.

## Abstract

An interplay between electron correlation and reduced dimensionality due to the Landau quantization gives rise to exotic electronic phases in three-dimensional semimetals under high magnetic field. Using an unbiased theoretical method, we clarify for the first time comprehensive ground-state phase diagrams of a three-dimensional semimetal with a pair of electron and hole pockets in the quantum limit. For the electron interaction, we consider either screened Coulomb repulsive interaction or an attractive electron-electron interaction mediated by a screened electron-phonon coupling, where a screening length is generally given by a dimensionless constant times magnetic length $l$. By solving the parquet RG equation numerically and employing a mean-field argument, we construct comprehensive ground-state phase diagrams of the semimetal in the quantum limit for these two cases, as a function of the Fermi wave length and the screening length (both normalized by $l$). In the repulsive interaction case, the ground state is either excitonic insulator (EI) in strong screening regime or Ising-type spin density wave in weak screening regime. In the attractive interaction case, the ground state is either EI that breaks the translational symmetries (strong screening regime), topological EI, charge Wigner crystal (intermediate screening regime), plain charge density wave or possible non-Fermi liquid (weak screening regime). We show that the topological EI supports a single copy of massless Dirac fermion at its side surface, and thereby exhibit a $\sqrt{H_{\perp}}$-type surface Shubnikov-de Haas (SdH) oscillation in in-plane surface transports as a function of a canted magnetic field $H_{\perp}$. Armed with these theoretical knowledge, we discuss implications of recent transport experiments on graphite under the high field.