Real-Space Spectral Approach to Orbital Magnetization

TL;DR

The paper tackles the challenge of computing orbital magnetization in crystals with disorder and realistic complexity, where k-space or projector-based real-space methods can be costly. It introduces a real-space spectral method based on the Chebyshev expansion of the operator $\hat{\mathscr{M}}_{z}\,\delta(E-\hat{H})$, defining the magnetization spectral density $m_z(E)$ and recovering $M_z(E_F)=\int^{E_F} m_z(E)\,dE$ without eigenstates. The approach uses open boundary conditions, linear scaling in system size, and stochastic traces to compute moments $\mu_n=\mathrm{Tr}[\hat{\mathscr{M}}_{z} T_n(\tilde{H})]$. Benchmarking on the Haldane model shows quantitative agreement with the modern k-space formula, captures edge contributions, and allows extraction of the Chern number from $m_z(E_F)$ via $-\frac{2\hbar c}{e}\,m_z(E_F)=\frac{\mathcal{C}}{2\pi}$; extending to disordered and defect-laden systems demonstrates robustness and scalability, making the method suitable for large, inhomogeneous quantum materials.

Abstract

We present a real-space spectral method for computing the orbital magnetization of crystals. Starting from the commutator form of the orbital magnetization operator, we formulate an energy-resolved spectral function that is amenable to exact Chebyshev polynomial expansions and yields the total magnetization upon integration up to the Fermi level. This avoids the need for computing eigenstates and ground-state projects, providing an efficient numerical framework that is applicable to very large systems even in the presence of disorder and temperature. Our approach is benchmarked on the Haldane model, finding results that are in excellent agreement with the modern $k$-space formulation of orbital magnetization. Leveraging this technique, we extend our study to systems with uncorrelated disorder and point defects, and further show that the bulk Chern number can be directly obtained from the magnetization spectral density. These results open a promising route to investigate orbital responses and topological transitions in real-space models of quantum materials with realistic complexity.

Figures (10)

• Figure 1: (a) Topological phase diagram showing the Chern number of the bottom band as a function of $\phi$ and $\Delta/t_{2}$ ($t_{1}=1$ eV and $t_{2}=t_{1}/3$). Red points $(\phi,\Delta/t_{2})=(0.5\pi,0)$, $(0.5\pi,6\sqrt{3})$ and $(0.1\pi,3)$ indicate the parameters for panels (b), (c) and (d), respectively. (b-d) Orbital magnetization (red lines) in real-space formulation and density of states (gray lines) as functions of the energy. The blue dots show the results of the reciprocal-space formulation. In our Chebyshev approach, we simulate a large rectangular domain with $10^{6}$ sites. Other parameters: $M=200$ and $R=800$.
• Figure 2: Disorder effects in Haldane model: (a-c) Orbital magnetization with Anderson disorder as functions of the Fermi energy $E_{F}$ for $W=1$ eV (red), $W=2$ eV (blue) and $W=3$ eV (green). The dashed gray lines correspond to the pristine systems. The panels (a), (b), and (c) refer to the red points in $(\phi,\Delta/t_{2})=(0.5\pi,0)$, $(0.5\pi,6\sqrt{3})$, and $(0.1\pi,3)$ of Fig. \ref{['fig:Fig_01']}(a).
• Figure 3: Chern number from the orbital magnetization: Panel (a) plots $-m_z$, calculated within the band gap for pristine systems at $\Delta/t_2=3\sqrt{3}\sin(\pi/4)$ and $\phi= 0.05\pi$, $0.10\pi$, $0.25\pi$, $0.30\pi$ and $0.40\pi$. The colored dots on each curve denote the respective CNP values. Other simulation parameters: $D\approx4\times10^6$, $M=200$ and $R=2000$. The inset represents the Chern number of the valence band of the Haldane model. The red points in this phase diagram are relevant for the subsequent discussion and figures. Panel (b) plots $-m_z$ at the CNP as a function of the sample sizes for pristine systems with $\phi= 0.20\pi$, $0.25\pi$ and $0.30\pi$. The panel (c) plots $-m_z$ at the CNP as functions of $\phi$ with $D\approx4\times10^6$, $M=4000$ and $R=2000$. The panel (d) plots $-m_z$ at the CNP as functions of $M$ for $W=1$ with $D\approx4\times10^6$ and $R=2000$.
• Figure 4: Point defects in the Haldane model: Panel (a) plots the orbital magnetization (red line) in real-space formulation and density of states (gray line) as functions of the energy, within the band gap for the topological insulating phase with $4\%$ vacancy concentration, $\phi=0.5\pi$ and $\Delta/t_2=0$. Panel (b) plots $-m_z$, calculated within the band gap for the same topological insulating phase. Other simulation parameters: $D\approx10^6$, $M=800$ and $R=2400$.
• Figure S1: (a-c) Orbital magnetization spectral density (red lines) in real-space formulation and density of states (gray lines) as functions of the energy. Panels (a), (b), and (c) show results for the systems with $(\phi,\Delta/t_{2})=(0.5\pi,0)$, $(0.5\pi,6\sqrt{3})$, and $(0.1\pi,3)$, respectively. Calculations are performed on a rectangular domain with $10^{6}$ sites. Chebyshev parameters: $M=200$ and $R=800$.