Continuum Landau surface states in a non-Hermitian Weyl semimetal

Abstract

The surface states of certain topological phases can be linked to a quantum anomaly: the violation of a classical symmetry by a field theory via a non-conserved current. This has been generalized to the case of a non-Hermitian (NH) chiral anomaly affecting the surfaces states of an NH Weyl phase. Here, we show that the NH anomaly inflow is mediated by continnum Landau modes (CLMs): special eigenstates exhibiting both spatial localization and a continuous spectrum, contrary to the usual distinction between bound and free states. Remarkably, the number of surface modes induced by the NH anomaly inflow scales linearly with the sample's volume, not surface area; this uniquely NH effect was not anticipated by earlier studies, and arises from the unusual multiplicity of the CLMs. The other properties of the CLMs, including their normalization conditions and localization scale, closely match the predictions of the NH field theory. Finally, we discuss the conditions under which these phenomena can be probed experimentally using metamaterials.

Figures (5)

• Figure 1: Surface states of a non-Hermitian Weyl semimetal with nonreciprocal couplings. (a) With periodic boundary conditions (PBC) along $y$, open boundary conditions (OBC) otherwise, and no external magnetic field, the $\pm x$ and $\pm z$ boundaries host surface states mapping to Hermitian Fermi arc surface states. (b) A magnetic field applied along $+z$ turns the NH surface states into "continuum Landau modes" (CLMs) on the $-z$ boundary. (c) With OBC in all directions, all eigenstates collapse into skin modes on the $\pm y$ boundaries. (d) When a magnetic field along $+z$ is further applied, the eigenstates are a mix of skin modes and CLMs.
• Figure 2: (a) 3D tight-binding model. Each unit cell contains two sites $A$ and $B$, with mass $\pm \Delta$. Adjacent cells are coupled by reciprocal hoppings $\pm t$ (black lines), nonreciprocal hoppings $\pm it$ (gray arrows; arrow direction indicates $+it$), and NH one-way hoppings (red arrows). In subsequent plots, we set $t = 1/2$ and $\Delta = 2$. (b) Complex spectrum for $y$-PBC and $B = 0$ [Fig. \ref{['fig:schematic']}(a)], with lattice size $L_x = L_z = 11$, $L_y = 22$. Colors denote the $z$ participation ratio (smaller means stronger localization). The boundary of the point gap, $|E| = 1$, is marked by red dashes. The states in $|E| < 1$ are NH Fermi arc surface states. (c) Complex spectrum for $y$-PBC and $\mathbf{B} = 2\pi/30\, \hat{z}$ [Fig. \ref{['fig:schematic']}(b)], with all other parameters the same as (b). The states in $|E| < 1$ are now CLMs on the $-z$ surface. (d) Surface state intensity $\sum |\psi|^2$ for a lattice of $L_x = L_z = 11$, with $y$-PBC. The intensity is summed over sites along the bottom ($-z$) surface, both sublattices, and all eigenstates with $|E| < 1$; results are shown for various $k_y$, with each curve receiving contributions from multiple states of different $E$. (e) Log-log plot of the spatial width $L$ of a CLM (with $E \approx 0$) versus magnetic field $B$ (blue dots). A least-squares fit of $\log L$ versus $\log B$ (red line) gives $L \propto 1/\sqrt{B}$wang2023continuum. All other parameters are the same as in (b). (f) Plot of $d\Delta n/dB \cdot (L_xL_yL_z)^{-1}$ versus $B$. Here, $\Delta n = n_{-}- n_{+}$ is the difference in the number of in-gap modes ($|E|<1$) localized on the bottom and top surfaces, and the derivative is estimated using $\delta B = 0.02$. Results are plotted using $L_x=L_y=11$, with different values of $L_z$. The yellow-shaded region indicates the regime $L_x \lesssim 2B^{-1/2}$, where the system size is smaller than the CLM width.
• Figure 3: (a) Complex spectrum under full OBC, with colors indicating the $z$-PR of the modes. (b) Complex spectrum under $B=0.4$. In-gap surface modes have a small $z$-PR as modes are pushed into the $-z$ plane. The point gap $|E|<1$ is indicated by the gray dashes. (c) Mode imbalance $\Delta n$ versus $B$ (blue dots), calculated numerically using eigenstates with $|E|<1$. Solid line shows the estimate $\Delta n = B L_x L_y L_z /2\pi$. (d) Mode intensity profiles for the representative eigenstates labeled in (b), which are localized to points on the $-z$ surface. The results for (a)--(d) are obtained with a $22\times22\times22$ lattice. (e) Number of modes with $|E|<1$ versus $L_z$, for $B=0.2$ and $L_x=L_y=11$ (blue dots). The blue line is the linear least-squares fit.
• Figure 4: (a) Transmittance spectrum on the $-z$ surface of a $y$-PBC sample, with source and probe on the sides of the $-z$ surface (inset schematic). For $B=0$ (blue line), transmission is nonzero over a range surrounding the point gap $|E|<1$ (yellow region). For $B=0.2$ (brown dashes), a strong peak develops in the point gap, corresponding to CLM surface states. Each curve is normalized to a maximum transmittance of 1. (b) Field intensity $I$ on the $-z$ surface for $B=0$ and $B=0.2$. The last four sites are excited by point sources. For $B=0$, the field decays exponentially away from the excitation (top plot), but for $B=0.2$ the decay is Gaussian. Solid lines show linear and quadratic fits of $\log I$ versus $x$, excluding the excitation points. These calculations use $y$-PBC with $k_y=0$ and $L_x=L_z=11$. (c) Intensity difference between bottom and top halves of the lattice, versus magnetic field $B$, for an OBC lattice with $L_x=L_y=21$ and $L_z = 11, \space 21$. The source is placed in the center of the lattice (see inset).
• Figure S1: (a) Numerical results illustrating the energy-position relation for CLM surface states of the NHWSM. For $y$-PBC lattices of size $L_x=L_y=11$, with different $L_z$, and applied magnetic field $B = 0.21$, we extract the eigenstates with $|E| < 1$, find $\langle k_x\rangle$ by Fourier transforming in $x$, and plot $\text{Re}E$ versus $\langle k_x\rangle$. (b) Distribution of CLM surface states, resolved by $k_y$. For $L_x = 11$ and $L_z = 22$, with two choices of $B$, we show the $x$-dependence of the surface intensity $\sum |\psi|^2$, where the sum is taken over sites on the bottom ($-z$) surface for all $|E| < 1$ eigenstates. With increasing $B$, more Gaussian wavepackets (corresponding to different $k_y$ fit in the sample, giving rise to an increase in the density of surface states; this is similar to Landau levels, but with a higher multiplicity of states for each $k_y$. (c) Variation of the wavepacket width versus $B$ in a full-OBC sample. The widths along $x$ and $y$, $\delta x$ and $\delta y$, are extracted from a representative surface state (initially identified at energy $E_0 = 0.155 - 0.063i$ for $B=0.4$) and tracked as $B$ varies. The system size is $L_x= L_y = L_z=11$, with OBC applied in all directions. The semilogarithmic best-fit curves, shown in red, agree with the predicted $B^{-1/2}$ scaling.