Breakdown of chiral anomaly and emergent phases in Weyl semimetals under orbital magnetic fields

TL;DR

This work shows that lattice effects fundamentally modify the magnetic-field-induced gap opening in Weyl semimetals. While continuum theories predict monotonic or oscillatory gap opening depending on anisotropy, the lattice introduces two distinct gap-opening channels (intra- and inter-BZ tunneling) and dispersive Landau levels, giving rise to NI, LCI, and, for certain anisotropies, LCI' phases. The authors map comprehensive phase diagrams for gamma<0 and gamma>0 using Bloch-Hofstadter theory, revealing exponentially narrow Weyl-semimetal windows and topology-driven surface-state evolution, including Fermi-arc behavior across transitions. These results have direct implications for transport and surface probes, offering Hall-geometry signatures to identify topological phases and highlighting the sensitivity of gap openings to field orientation and lattice periodicity.

Abstract

An external orbital magnetic field applied perpendicular to the separation vector of a pair of Weyl points can couple them and induce a gap in the electronic spectrum. In this work, we investigate the gap-opening behavior in the presence of a lattice, revealing rich phenomenology absent in the continuum picture. Specifically, we address the emergence of layered Chern insulating states, examining how the anisotropy of the Weyl cone dispersion influences the sequence of phase transitions, and establishing connections to the continuum limit. We analyze the evolution of surface Fermi-arc states across these regimes, highlighting their distinct behaviors during the gap-opening transitions.

Figures (9)

• Figure 1: Schematic of a Weyl semimetl with two Weyl cones placed in a uniform magnetic field ${\bf B} \parallel \hat{z}$. Weyl cones are separated along $k_x$ and their momentum space separation is $Q$. Magnetic field is aligned perpendicular to the direction of separation between Weyl cones of opposite chirality. The orbital field ${\bf B}$ introduces a quantum tunneling of particles between the Weyl cones which can induce a finite gap in the spectrum. A schematic phase diagram is depicted in Fig. \ref{['Fig:Phase_Schematic']}.
• Figure 2: Band structures for $\gamma<0$ and $\gamma>0$ are shown in (a1) and (b1) respectively. (a2)-(b2) Constant energy contours around the Weyl nodes have elliptical and crescent shape for $\gamma<0$ and $\gamma>0$ respectively. Weyl nodes are represented by black dots. There is only a single close encounter for $\gamma<0$, and two close encounters for $\gamma>0$ between the Fermi pockets around the Weyl nodes. (a3)-(b3) Schematic phase diagram (for a system on a lattice) in presence of an orbital magnetic field ${\bf B} \parallel \hat{z}$ which is perpendicular to the direction of separation between two Weyl nodes of opposite chirality. Here $Q$ and $Q'$ represent the intra and inter BZ separation between the Weyl nodes in momentum space. (a3) For $\gamma<0$, the system displays three phases as a function of the separation between the Weyl nodes $Q$. The transition from normal insulator (NI) to layered Chern insulator (LCI) passes through a gapless WSM phase. The width of the WSM region is exponentially small in $\phi_B^{-1}$. (b3) For $\gamma>0$ and a fixed applied flux, there is a series of phase transitions as we tune the separation $Q \in(0, 2\pi)$. Besides NI and LCI, another insulating state which we denote as LCI$'$ appears. The state LCI$'$ can be viewed as two copies of LCI with opposite Chern numbers. All transitions between NI and LCI$'$ phases pass through a sequence of three phases: two WSMs and an LCI.
• Figure 3: Energy gap ($\Delta$) in the continuum model as a function of the square of the dimensionless parameter $Ql_B$, computed numerically for $t=1$. (a) For $\gamma=-0.44$, the field-induced gap falls exponentially with $Ql_B$. (b) For $\gamma=0.96$, the gap oscillates and periodically goes to zero (at the troughs). The troughs in the figure are not exactly at zero because of the finite line grid in the numerical computation. Here the envelope of oscillations falls exponentially.
• Figure 4: Energy spectrum (along $k_x$) in a slab geometry along $z$ direction with thickness $L_z=50$ in units of lattice constant for $\gamma \approx -0.30$ (negative). Energies are given in units of $v_x$. Upper panels show the spectrum for zero field, highlighting the zero energy Fermi-arc surface states (a) for small ($Q=1$) and (b) large ($Q=5$) separation between the Weyl nodes. Bottom panels show the spectrum in presence of an external field applied along $z$ direction corresponding to a flux of $\phi_B/\phi_0=1/q$ with $q=10$. (c) Fermi-arc states get fully gapped out (NI phase). (d) Fermi-arc states (highlighted in red) are extended to the edge of the magnetic BZ along $k_x$ (LCI phase).
• Figure 5: Behavior of the gap for $\gamma<0$. (a) Zero field energy spectrum is showing two Weyl cones separated along $k_x$. (b) In presence of an orbital ${\bf B}\parallel \hat{z}$, Weyl cones are gapped out. Only two Landau bands near zero energy are shown. For flux $\phi_B \sim \phi_0$, the bands have a finite bandwidth ($W$). (c) Energy gap ($\Delta$) as a function of Weyl nodes' separation for flux values $\phi_B/\phi_0 = 1/q=1/10, 1/20, 1/50$. Magnetic field immediately creates a gap in the spectrum for almost any value of separation between the Weyl nodes. The gap is exponentially small $\sim \exp(-Q^2l_B^2)$. For intermediate values of separation, WSM state survives but in an exponentially small vicinity of $k_0$ (seen most clearly for $\phi_B=1/3$). (d) The width of the gapless region (denoted by $\Delta Q$ in (c)) where the system remains in the WSM phase falls exponentially with $\phi_B^{-1}$. The bandwidth (denoted as $W$ in (b)) also falls as $\exp(-\phi_B^{-1})$.