Abstract
Skin effect, experimentally discovered in one dimension, describes the physical phenomenon that on an open chain, an extensive number of eigenstates of a nonHermitian Hamiltonian are localized at the end(s) of the chain. Here in two and higher dimensions, we establish a theorem that the skin effect exists, if and only if periodicboundary spectrum of the Hamiltonian covers a finite area on the complex plane. This theorem establishes the universality of the effect, because the above condition is satisfied in almost every generic nonHermitian Hamiltonian, and, unlike in one dimension, is compatible with all pointgroup symmetries. We propose two new types of skin effect in two and higher dimensions: the cornerskin effect where all eigenstates are localized at corners of the system, and the geometrydependentskin effect where skin modes disappear for systems of a particular shape, but appear on generic polygons. An immediate corollary of our theorem is that any nonHermitian system having exceptional points (lines) in two (three) dimensions exhibits skin effect, making this phenomenon accessible to experiments in photonic crystals, Weyl semimetals, and Kondo insulators.
Introduction
The study of nonHermitian Hamiltonians, which can be regarded as the effective description of dissipative processes, can be traced back to the investigation of alpha decay, where real and imaginary parts of the complex energy are related to the experimentally observed energy level and decay rate^{1}. When a lattice system is coupled with environments and has dissipations, e.g. photonic crystals having radiational loss^{2,3,4} and electronic systems having finite quasiparticle lifetime^{5,6}, the nonHermitian band theory becomes a conceptually simple and efficient approach^{7,8,9,10,11,12}.
Skin effect^{13,14,15,16,17,18,19,20,21,22,23}, a phenomenon unique to the nonHermitian band theory, refers to the localization of eigenstates at the boundary, the number of which scales with the volume of the system. For example, in one dimension, all eigenstates of a nonHermitian Hamiltonian can be localized at the ends of a chain^{13}. This suggests the failure of Bloch’s theorem^{24,25}, which states that eigenstates in the bulk are modulated plane waves. As Bloch’s theorem plays a fundamental role in the development of condensedmatter physics^{26}, the emergence of skin effect indicates a new and possibly revolutionary direction. Especially, the skin effect has been observed experimentally in onedimensional classical systems^{27,28,29}, inspiring further studies on their higher dimensional generalizations^{14,30,31,32,33,34,35,36,37}. However, a general theory for the higherdimensional skin effect has not been established.
Apart from the skin effect, another focus topic in nonHermitian band systems is the exceptional point (or line)^{38,39,40,41,42,43,44,45,46,47} that refers to stable pointtype (or linetype) nonHermitian band degeneracy in the Brillouin zone. At the exceptional point, not only eigenvalues but also eigenstates of the Bloch Hamiltonian coalesce^{39}. Many a novel phenomenon related to exceptional points has been predicted and observed^{47,48,49,50,51,52}, such as the emergence of bulkFermi arc terminated at the exceptional points^{5,45}. Since the bulkboundary correspondence plays a central role in the development of topological phases^{53}, it is natural to ask if there exists a nonHermitian bulkboundary correspondence in bands having exceptional points, analogous to the surface Fermi arc in the Weyl semimetals in the Hermitian counterpart^{54}.
In this paper, we establish a theorem that reveals a universal bulkboundary correspondence in two and higher dimensional nonHermitian bands, as shown in Fig. 1. The “bulk” refers to the area of the spectrum of the Hamiltonian on the complex plane with periodic boundary condition, and “boundary” means the presence (absence) of the skin effect for openboundary system of a generic shape. The theorem states that the skin effect appears if and only if the spectral area is nonzero. This skin effect is “universal” for three reasons: (i) a randomly generated local nonHermitian Hamiltonian has the skin effect with probability one; (ii) the skin effect is, unlike in one dimension, compatible with all pointgroup symmetries and timereversal symmetry, including complexconjugatetype and transposetype timereversal symmetry in ref. ^{11}; and (iii) it does not require any special geometry of the openboundary system. We classify the universal skin effect into nonreciprocal skin effect and generalized reciprocal skin effect according to nonzero and zero current functional, respectively, and also propose the cornerskin effect and geometrydependentskin effect as representative phenomena of these two categories.
A surprising corollary of our theorem is that the stable exceptional points^{8,41,43} imply the presence of skin effect. Because exceptional points have been either observed or proposed in metamaterials as well as in condensed matter, this corollary makes skin effect observable in known systems. We predict the geometrydependent skin effect in the twodimensional photonic crystal studied in ref. ^{45}, and propose to observe this effect in the anomalous dynamics of wave packets.
Results
Theorem: an equivalence between spectral area and skin effect
For generic onedimensional nonHermitian systems, the correspondence between the spectral shape and the skin effect has been derived^{17,18}, i.e., when the Bloch spectrum is a looptype (an arctype), the skin effect appears (disappears).
Generalizing the correspondence to two dimensions, we note two main differences. One difference is in the periodicboundary spectrum, E_{i}(k), where i is the band index and k the crystal momentum in the first Brillouin zone (BZ). Generally speaking, E_{i}(k) is a mapping from the ddimensional torus to the complex plane, \({\mathbb{C}}\). When d = 1, the image of E_{i}(k) forms a loop; but when d > 1, the image is generically a continuum on \({\mathbb{C}}\), denoted by E_{i}(BZ). The area covered by E_{i}(BZ) on the complex plane is called the spectral area, denoted by A_{i}. Another difference is in the variety of openboundary condition. There is only one geometry for an open system in one dimension, i.e., an open chain; but there are an infinite number of geometries in two dimensions such as triangle, rectangle and pentagon.
Now we are ready to state the theorem of universal skin effect: in the thermodynamic limit, the skin effect is present in a Hamiltonian having open boundary of generic geometry, if the spectral area is nonzero (A_{i} ≠ 0); vice versa, the skin effect is absent for all possible geometries, if the spectral area is zero (A_{i} = 0). The open boundary in the theorem refers to the fully open boundary condition in all spatial directions. As the periodicboundary Hamiltonian describes the dynamics in the bulk, the theorem relates a bulk property (spectral area) to a boundary one (existence of skin modes). Fig. 1 shows some schematic examples. The complete proof of the theorem is provided in the Supplementary Note 1.
A brief outline of the proof is illustrated in Fig. 2. The theorem is obtained in three steps: step I establishes the equivalence relation between spectral area and spectral winding number of straight lines in the BZ; step II connects these nonzero spectral winding numbers with skin effect on the stripe geometry — the geometry with open boundary in only one direction and periodic boundary in other directions; step III illustrates that skin effect on stripe geometry implies skin effect on fully openboundary geometry (i.e., the universal skin effect), which relies on a conjecture. The justification of this conjecture is discussed in the Supplementary Note 1.
The above theorem has implied the universality of skin effect in two and higher dimensions. As E_{i}(BZ) is the image of the d ≥ 2dimensional torus on the complex plane, it takes fine tuning of parameters to make A_{i} = 0 for every i. In fact, for singleband Hamiltonian, we can prove that A = 0 if and only if \({{{{{{{\mathcal{H}}}}}}}}({{{{{{{\bf{k}}}}}}}})=P[h({{{{{{{\bf{k}}}}}}}})]\), where h(k) is a Hermitian Hamiltonian and P is a complex polynomial (see Supplementary Note 1). In other words, a randomly generated nonHermitian Hamiltonian \({{{{{{{\mathcal{H}}}}}}}}({{{{{{{\bf{k}}}}}}}})\) has skin effect: the first meaning of universality. In previous studies, other types of skin effect, such as the lineskin and the highorderskin effect, in two and higher dimensions have been proposed^{30,34}. These types all require the openboundary system take a special geometry (usually a rectangle) and are hence considered special and nongeneric. Additionally, the number of skin modes in the universal skin effect follows a volume law, which differentiates from the higherorderskin effect. The skin effect when A_{i} ≠ 0 assumes a completely generic geometry of boundary: the second meaning of universality. The third meaning of universality lies in the fact that the higherdimensional skin effect is compatible with all pointgroup symmetries, i.e., the universal skin effect can appear if and only if the spectral area is nonzero, regardless of the pointgroup symmetry of the bulk Hamiltonian. While in one dimension, if the bulk Hamiltonian only respects, for example, the inversion symmetry, the periodicboundary energy spectrum has an arcform on the complex plane, which means the absence of nonHermitian skin effect^{11,23}. A standing wave explanation for the theorem is provided in the Supplementary Note 2.
The cornerskin and the geometrydependentskin effect
While the theorem shows that the skin effect is universal, it does not specify what skin modes look like in higher dimensions. Here, we define the current functional, which partitions the universal skin effect into nonreciprocal skin effect and generalized reciprocal skin effect. Then we report the representative phenomena in these two types, i.e., the cornerskin effect (CSE) and the geometrydependent skin effect (GDSE), respectively.
The current functional is defined as
under the periodicboundary condition, where i is the energy band index and ∇_{α} indicates the directional derivative along certain direction α in ddimensional momentum space. Here, n(E, E^{*}) represents a distribution function when the system reaches to a steady state and only depends explicitly on the energy of the system state^{17}. The nonzero current functional (labeled by J ≠ 0) is defined as: ∃ α, n, J_{α}[n] ≠ 0; and as a complementary set, the zero current functional (labeled by J = 0) is defined as: ∀ α, n, J_{α}[n] = 0. By definition, the nonzero current functional and zero current functional are complete and mutually exclusive mathematically. Therefore, we can classify the universal skin effect (nonzero spectral area) into two types according to the current functional, i.e., the nonreciprocal skin effect (J ≠ 0) and generalized reciprocal skin effect (J = 0), as illustrated in Fig. 3. Note that this classification of skin effect according to the current functional is different from the classification of intrinsic pointgap topology for symmetry class^{11,18} (see Supplementary Note 3). The current functional is shown to vanish in two and three dimensions under point groups C_{i}, D_{2,3,4,6}, C_{2h,3h,4h,6h}, D_{2d,3d,2h,3h,4h,6h}, T, T_{d,h}, O and O_{h}. Therefore, the nonreciprocal skin effect is only compatible with point groups C_{m} and C_{2,3,4,6,2v,3v,4v,6v}. As a comparison, the generalized reciprocal skin effect is compatible with all pointgroup symmetries (see also Supplementary Note 3).
We define the CSE as a type of the nonreciprocal skin effect (J ≠ 0) that exhibits the particular phenomenon that almost all eigenstates are localized at corners of the openboundary geometry. The Hamiltonian of the example for CSE is
of which the spectral area under square geometry and triangle geometry is shown in Fig. 3(a, b) with light blue color. Because of the nonzero spectral area, the theorem tells us that the Hamiltonian must have the universal skin effect. This is verified in Fig. 3(c, d), where the spatial distributions of all eigenstates \(W({{{{{{{\bf{x}}}}}}}})=\frac{1}{N}{\sum }_{n}{\left{\psi }_{n}({{{{{{{\bf{x}}}}}}}})\right}^{2}\) under different open boundaries are plotted. Here ψ_{n}(x) is a normalized right eigenstate and N is the number of these eigenstates. It is found that the wave functions are always localized at the corner of the boundary in Fig. 3(c), even if the openboundary geometry is changed in Fig. 3(d). We elaborate on the localization of eigenstates for this example in the Supplementary Note 4. We also plot the corresponding eigenvalue spectra under different open boundaries, as shown in Fig. 3(a, b) with red color. One can notice that the spectral areas under periodic and open boundaries do not equal. The CSE is a representative one of nonreciprocal skin effect and inherits its features, including nonzero current functional and incompatibility with certain pointgroup symmetries.
Similar to the definition of CSE, the GDSE is one type of generalized reciprocal skin effect (J = 0) showing the unique phenomenon that there is at least one fully open boundary geometry under which the skin effect does not appear. The Hamiltonian of the example for GDSE reads
Since the spectral area is nonzero, our theorem tells us that the system must have skin effect for certain openboundary geometry, such as a random polygon. However, an interesting phenomenon in this example is that the skin effect disappears under the square geometry due to the existence of two mirror symmetries shown in Fig. 3(g). Once we choose other types of boundaries where mirror symmetries are broken, the skin effect reappears as shown in Fig. 3(h). Since the appearance of the skin effect and the localization position depend on the geometry, it is called the GDSE. In one dimension, an open chain does not exhibit skin effect when its spectrum coincides with the corresponding periodicboundary spectrum on the complex plane. Unlike in one dimension, even if the region covered by the energy spectrum under some openboundary geometry (such as the triangle geometry in Fig. 3(h)) seems to be the same as the region covered by the periodicboundary spectrum, the system can still show a skin effect due to the different density of states on the complex plane. It is also a unique feature in two and higherdimensional skin effects. In the Supplementary Note 4, we provide some numerical results to illuminate this new type of skin effect and discuss the localization of eigenstates on the openboundary geometry. In addition, we show that GDSE follows the volume law, i.e., the increase in the number of skin modes is proportional to the increase in the system volume. For GDSE, there is at least one spatial geometry such that skin modes vanish, and as such is mutually exclusive with CSE. Additionally, GDSE is compatible with all point groups, in contrast to CSE.
Corollary: skin effect from exceptional points
An immediate corollary of our theorem is that all lattice Hamiltonians with stable exceptional points have universal skin effect, connecting two unique phenomena in the nonHermitian band theory. This connection has also been discussed in ref. ^{31}, where the bands around the stable exceptional point form a point gap with nonzero spectral winding number, consequently, exhibiting the skin effect under an openboundary geometry. Consider a stable exceptional point k_{0} in two dimensions. Due to the branch point structure of exceptional point, the dispersion around k_{0} can be expressed as^{8}\({E}_{\pm }({{{{{{{\bf{k}}}}}}}})=\pm {c}_{0}\sqrt{{q}_{x}+{c}_{1}{q}_{y}}+O( {{{{{{{\bf{k}}}}}}}}{{{{{{{{\bf{k}}}}}}}}}_{0} )\), where q_{i=x,y} denotes a small derivation from exceptional point in x or y direction, that is, q_{i} = k_{i} − k_{0i}. Here c_{0}, c_{1} are nonzero complex numbers and the stable exceptional point ensures the nonzero imaginary part of c_{1}. Suppose the range of the expansion is r_{0}, then it is clear that \({A}_{\pm }\ge  {c}_{0} \pi {{r}_{0}}^{2}/2\,\ne\, 0\). By the theorem, the system must have universal skin effect (see Supplementary Note 5).
Now we use the photonic crystal model that has been experimentally realized in ref. ^{45} to demonstrate our corollary. The tightbinding model Hamiltonian with periodic boundary can be written as
where σ = (σ_{0}, σ_{x}, σ_{y}, σ_{z}) is a vector of the Pauli matrices and d(k) is a vector with four components, that is, \({{{{{{{\bf{d}}}}}}}}({{{{{{{\bf{k}}}}}}}})=\{{\mu }_{0}({t}_{2}+{t}_{3})(\cos {k}_{x}+\cos {k}_{y}),{t}_{1}[1\cos {k}_{x}\cos {k}_{y}+\cos ({k}_{x}{k}_{y})],{t}_{1}[\sin {k}_{x}\sin {k}_{y}\sin ({k}_{x}{k}_{y})],{\mu }_{z}+({t}_{2}{t}_{3})(\cos {k}_{x}\cos {k}_{y})\}\). The parameters are chosen as follows, (t_{1}, t_{2}, t_{3}, μ_{0}, μ_{z}) = (0.4, − 0.1, 0.5, 1.35, − 0.02). As shown in Fig. 4(a), in the Hermitian limit, i.e. γ = 0, the system has two Dirac points along the xaxis. When external dissipation or radiational loss is added, i.e., γ ≠ 0, each Dirac point splits into two exceptional points shown in Fig. 4(b), connected by the bulk Fermi arc. According to our theorem, the system must have the universal skin effect, more precisely, the GDSE. Specifically, the skin effect disappears under square geometry but reappears under diamond geometry, which is verified in the Supplementary Note 6.
So far, we have shown the features of the energy spectrum and wave function in the system with GDSE. We expect some observable phenomena from the skin effect, which motivates us to examine the dynamical properties for the photonic crystal model in Eq.(4). In order to show this, we simulate the time evolution of the wave packet starting at the center of the diamond geometry with an initial velocity perpendicular to one edge. Here the initial state is chosen to be Gaussian form \(\left{\psi }_{0}\right\rangle ={{{{{{{\mathcal{N}}}}}}}}\exp [{\left(x{x}_{0}\right)}^{2}/10{\left(y{y}_{0}\right)}^{2}/10i2xi2y]{\left(1,1\right)}^{T}\), where \({{{{{{{\mathcal{N}}}}}}}}\) is the normalization factor and x_{0} = y_{0} = 21 is the center coordinate of the diamond geometry. We plot the corresponding spatial distribution of normalized final states \(\left\psi ({t}_{f})\right\rangle ={{{{{{{\mathcal{N}}}}}}}}({t}_{f}){e}^{i{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{OBC}}}}}}}}}{t}_{f}}\left{\psi }_{0}\right\rangle\) for every ten time intervals, where \({{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{OBC}}}}}}}}}\) represents the openboundary Hamiltonian on the diamond geometry. As shown in Fig. 4(c), in the Hermitian case, the center of the wave packets obeys the simple law of reflection: the center of the wave packet just bounces between the two edges while slowly dispersing with time. However, in the nonHermitian case (γ = 1/4) with GDSE, after several oscillations between two edges, the wave packet makes a side jump into the upper left corner as shown in Fig. 4(d). The transverse motion of the wave packet induced by skin effect is explained in more detail in the Supplementary Note 6. This anomalous dynamical behavior is an experimental signature of GDSE.
We also propose the realization for CSE in a threedimensional system with exceptional lines. Consider a Weyl semimetal with nonHermitian term as a perturbation, of which the periodicboundary Hamiltonian reads
where d_{r}(k) and d_{i}(k) are vectors with four components, that is, \({{{{{{{{\bf{d}}}}}}}}}_{r}({{{{{{{\bf{k}}}}}}}})=(0,\sin {k}_{x},\sin {k}_{y},2\cos {k}_{x}\cos {k}_{y}+\sin {k}_{z})\) and \({{{{{{{{\bf{d}}}}}}}}}_{i}({{{{{{{\bf{k}}}}}}}})=(\sqrt{5},1+\cos {k}_{z},1\cos {k}_{z},\cos {k}_{z})\). The Hermitian part d_{r} ⋅ σ is a Weyl semimetal possessing two Weyl points, the red cone at (0, 0, 0) and blue cone at (0, 0, π) shown in Fig. 4(e). Upon turning on the nonHermitian term, the Weyl points evolve into two exceptional rings in Fig. 4(f). Consequently, the system exhibits the CSE with δ = 1/6 shown in Fig. 4(g), as a numerical verification of our corollary.
Experimentally, the nonreciprocity of the CSE can be detected by the twopoint Green’s function. The modulus square of the propagator from i = (1, 1, 1) to o = (16, 16, 16) is expressed as \({P}_{oi}(\omega )={\sum }_{\alpha ,\beta }{\langle o,\beta \frac{1}{\omega \hat{H}}i,\alpha \rangle }^{2}\), where α, β label the orbitals of the unit cell. We calculate P_{oi}(ω) and P_{io}(ω) in Fig. 4(h), where a significant difference between them demonstrates the nonreciprocity of CSE.
Discussion
Our work has built a bridge between two distinct phenomena that only exist in nonHermitian systems, i.e., the exceptional points (lines) and the nonHermitian skin effect, by establishing the correspondence between bulk (spectral area) and boundary (universal skin effect). We prove that the skin effect is universal and compatible with all pointgroup symmetries and timereversal symmetry in two and higher dimensions. Due to the universality, it is expected that the skin effect is observable in a wide range of platforms, such as photonic crystals with natural radiational loss, acoustic metamaterials and circuit networks with lossy components such as resistors. Beyond these classical systems, the skin effect can also be realized in condensed matter, e.g., the heavyfermion material with finite quasiparticle lifetime and the Weylexceptionalring semimetal. The latter is realizable in Weyl semimetals made from inverting bands that have disparate effective masses, such as d and fbands.
One should be reminded, however, that the results in this paper assume the coherent dynamics of the constituent degrees of freedom, which is unlikely the case in macroscopic condensedmatter systems where the coherence length is shorter than the system size. On the contrary, for the systems where the system size and the coherent length are comparable, as in mesoscopic systems, we believe that the universal skin effect has a significant contribution to the transport properties, a subject for future exploration.
Data availability
Raw numerical data from the plots presented are available from the authors upon request.
Code availability
The code used to generate the figures are available from the authors upon reasonable request.
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Acknowledgements
C.F. acknowledges funding support by the Ministry of Science and Technology of China under grant number 2016YFA0302400, and the Chinese Academy of Sciences under grant number XDB33000000. Z.Y. acknowledges funding support by the National Science Foundation of China (Grant No. NSFC12104450) and the fellowship of China National Postdoctoral Program for Innovative Talents (Grant No. BX2021300).
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C.F. conceived the work; K.Z. did the major part of the theoretical derivation and numerical calculation; Z.Y. wrote and analyzed the tightbinding Hamiltonian of the photonic crystal model; All authors discussed the results and participated in the writing of the manuscript.
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Zhang, K., Yang, Z. & Fang, C. Universal nonHermitian skin effect in two and higher dimensions. Nat Commun 13, 2496 (2022). https://doi.org/10.1038/s41467022301616
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DOI: https://doi.org/10.1038/s41467022301616
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