Topological chiral-gain in a Berry dipole material

2.1 Band structure and bulk stability

In this article, we focus on the propagation in the plane perpendicular to the trigonal axis of the material (xoy plane). The plane wave modes supported by the material are decoupled into transverse electric (TE) waves, with E _z ≠ 0, and transverse magnetic (TM) waves, with H _z ≠ 0. Evidently, only the TM waves ( E = E x x ̂ + E y y ̂ , H = H z z ̂ , and ∂/∂z = 0) are sensitive to the electrically induced gyrotropy. Therefore, our analysis is centered on this case. The dispersion of these waves is governed by the characteristic equation [9]:

being k = k x x ̂ + k y y ̂ the wave vector, k ⋅ k = k 2 = k x 2 + k y 2 and ε ef = ε D 2 − ε EO 2 / ε D .

To evaluate the impact of the electric bias on the material’s band structure and the stability of its optical response, we next examine the projected band structure for wave propagation in the xoy plane. The projected band structure represents the complex eigenfrequencies of the bulk modes for real-valued wave vectors. These eigenfrequencies are determined by solving Eq. (4) with respect to ω = ω′ + iω″.

In the first example, in Figure 2a we plot the projected band structure for the case where there is no bias (ω _c = 0) corresponding to a material described by a dissipative Drude metal, with Γ = 0.5ω _p. Due to the dissipation effects, the complex spectrum is completely confined in the lower-half frequency plane (ω″ < 0). Note that we represent both the positive (ω′ > 0) and negative (ω′ < 0) parts of the frequency spectrum, which are linked by the particle-hole symmetry ω → −ω*. Additionally, a quasi-static branch of modes appears along the imaginary axis (ω′ = 0).

Figure 2:

Projected band structure of the Berry-dipole material under the influence of an applied static electric field. The plots show a parametric representation of the complex-frequencies ω = ω′ + iω″ as a function of the real-valued wave vector in the xoy plane. The collision frequency is Γ = 0.5ω _p . The black dots represent the limiting points for k → ∞, while the blue dots represent the limiting points for k = 0. (a) ω _c = 0 (without the applied static electric field). (b) ω _c = 0.1ω _p, (c) ω _c = ω _c,th ≈ 0.5ω _p. The shaded vertical strip represents the relevant band gap, with gap Chern number −1.

In Figure 2b and c, we illustrate the impact of the bias on the band structure of the material, for a weak and strong bias, respectively. The bias induces the opening of a band gap, as indicated by the two vertical pink-shaded strips. Specifically, the bias separates the original positive-frequency branch of the Drude model into two distinct components. In the limit of a weak bias, ω _c → 0 the two branches become the standard transverse and longitudinal bands of the unbiased plasma. The limiting points of each branch are marked in the dispersion curves with blue dots (k = 0) and black dots (k → ∞). Note that for a trivial bias (Figure 2a), some of the blue and black dots coincide because the longitudinal plasmons have a dispersion independent of k. The topological charge of the band gap will be characterized in the next section.

Next, we investigate the conditions required to ensure a stable bulk response for wave propagation in the xoy plane. The material response is stable when the natural modes are in the lower-half of the complex frequency plane ω = ω′ + iω″. We employ an approach similar to that used in Ref. [48], which analyzed the stability for propagation along the trigonal axis.

Specifically, we note that at the instability threshold, ω _c,th, there must be an eigenmode with real valued frequency ω and real-valued wave vector k. In other words, the band structure must “touch” the real-frequency axis. Thereby from Eq. (4), it follows that Im ε ef ω = 0 . Solving this equation with respect to ω _c we find that:

The above formula gives the required bias strength so that the dispersion diagram intersects the real-frequency axis at frequency ω. The instability threshold ω _c,th, is found by minimizing Eq. (5) with respect to ω. Differentiating Eq. (5) with respect to ω and setting the result to zero, one finds that:

The frequency ω _las gives the lasing frequency of the bulk material at the instability threshold. Figure 3a represents the bias strength (Eq. (5)) required to have an intersection at the frequency ω. The minimum of the plot occurs at the lasing frequency ω _las. The plot assumes Γ = 0.5ω _p. For relatively small Γ/ω _p, the lasing frequency is roughly ω las ≈ ω p 9 2 + 4 ω p 2 Γ 2 > 2.12 ω p . Thus, the optical gain always arises near a frequency where the material behaves as a dielectric ( Re ε D > 0 ).

Figure 3:

Study of the stability of the optical response. (a) Bias strength ω c required to have a crossing at the real frequency ω (Eq. (5)). The collision frequency is Γ = 0.5ω _p. The black dot marks the lasing frequency ω _las for ω c = ω c,th (Eq. (6)). (b) Cyclotron frequency at the stability threshold as a function of the collision frequency Γ. (c) Projected band structure for a material with Γ = 0.5ω _p and (i) ω _c = ±0.5ω _c,th, (ii) ω _c = ±ω _c,th, and (iii) ω _c = ±1.5ω _c,th. The orange points mark the lasing frequency ω _las at the instability threshold. (d) Eigenvalues of the non-Hermitian part of the material response ε ̄ ′ ′ = ε ̄ − ε ̄ † / 2 i , for ω _c = 0, ω _c = 0.25ω _p, and ω _c = 0.75ω _p.

Evidently, the minimum of ω _c occurs for ω c,th = ω c ω las . Substituting Eq. (6) into (5), we find that:

The material response is stable provided ω c < ω c,th . It is clear from the previous equation that for small Γ/ω _p, the equivalent cyclotron frequency at the instability threshold is ω _c,th ≃ 0.5ω _p. As shown in Figure 3b, this estimate remains quite accurate even for fairly large values of Γ/ω _p.

Figure 3c shows the projected band diagram of the material, for the cases of a stable bulk response ( ω c = 0.5 ω c,th ), at the instability threshold ( ω c = ω c,th ), and for an unstable response ( ω c = 1.5 ω c,th ), for the normalized collision frequency Γ = 0.5ω _p. As seen, the response is stable for ω c ≤ ω c,th and the dispersion diagram crosses the real frequency axis nearby the lasing frequency ω _las (orange points).

2.2 Non-Hermitian response

For non-conservative materials, it is useful to decompose the permittivity tensor as ε ̄ = ε ̄ ′ + i ε ̄ ′ ′ , where ε ̄ ′ = ε ̄ + ε ̄ † / 2 and ε ̄ ′ ′ = ε ̄ − ε ̄ † / 2 i are both Hermitian tensors. The tensor ε ̄ ′ governs the conservative part of the material response and determines the isofrequency contours of the bulk modes. In contrast, ε ̄ ′ ′ describes the power dissipation in the material per unit of volume, given by p d = 1 2 ω ε 0 E * ⋅ ε ̄ ′ ′ ⋅ E . In conventional systems, the light–matter interactions are invariably dissipative, meaning that p _d > 0. In this case, the tensor ε ̄ ′ ′ must be positive definite, with strictly positive eigenvalues. However, in systems with optical gain, the power p _d can be negative because the material can supply energy to the wave.

A straightforward analysis shows that, for the optical response defined by Eqs. (2) and (3), the eigenvalues of the tensor ε ̄ ′ ′ take the form:

The corresponding eigenvectors are the two circular polarizations e ̂ + = 1 2 x ̂ + i y ̂ and e ̂ − = 1 2 x ̂ − i y ̂ . There is also an additional eigenvector e ̂ ∼ z ̂ , corresponding to an electric field oriented along the z-axis, but it is omitted here as we focus on TM waves, where the electric field is confined to the xoy plane.

As seen from Eq. (8), in the absence of the static electric bias (ω _c = 0), the two eigenvalues are coincident and positive (blue line in Figure 3d), typical of a passive response. As usual, the dissipation is rooted in the collisions of free-electrons with the ionic lattice.

Interestingly, under a nontrivial static bias (ω _c ≠ 0), the eigenvalues ε + ′ ′ , ε − ′ ′ acquire an additional contribution. This corresponds to the second terms in Eq. (8), which arise from the linear electrooptic effect, and have opposite signs. Thereby, as the static bias strength increases, one of the circular polarizations experiences greater dissipation, while the opposite-handed circular polarization experiences reduced dissipation (see Figure 3d, case ω _c = 0.25ω _p). Remarkably, for a strong enough bias one of the eigenvalues can become negative, corresponding to a situation where the material can experience gain. This is illustrated in Figure 3d, for ω _c = 0.75ω _p. In this case, one of the eigenvalues ( ε − ′ ′ ) becomes negative (dashed black curve) while the other one ( ε + ′ ′ ) remains positive (solid black curve). This means that one of the circular polarizations ( e ̂ + ) originates dissipation, while the circular polarization with opposite handedness ( e ̂ − ) originates gain. This effect defines the material as a chiral gain medium. It is relevant to note that when the static bias is flipped, so that ω _c becomes negative, the eigenpolarizations that activate the gain and dissipation are interchanged. It is important to emphasize that eigenpolarizations e ̂ ± are intrinsic properties of the bulk material and bias, independent of the specific electromagnetic field distribution, how it is generated, or the direction of propagation. Excitations or material geometries that favor fields with circular polarizations lead to the strongest non-Hermitian effects, such as enhanced gain (or dissipation) in the case of e ̂ − e ̂ + .

It is useful to introduce the spin angular momentum of the electric field, defined as σ = i E × E * / E 2 [56]. The spin angular momentum is controlled by the handedness of the polarization curve. The spin angular momentum of the e ̂ ± eigenfunctions is σ = ± z ̂ .

The part of the dissipated power p _d associated with the non-Hermitian linear electrooptic effect can be written in terms of σ as:

We refer to the dimensionless vector Ω _ω as the “gain vector”. The formula above indicates that gain (or dissipative) interactions occur when the spin angular momentum of the wave is parallel (or anti-parallel) to the gain vector. This result generalizes the findings reported in Ref. [57] for dispersionless systems.

From Eq. (8), it is evident that for a chiral-chain configuration, where the eigenvalues ε + ′ ′ , ε − ′ ′ have opposite signs, the bias strength must satisfy ω c > ω p 2 / ω within a certain frequency range. This condition is fully compatible with the requirement for bulk stability ( ω c ≤ ω c,th ), because for large frequencies ω p 2 / ω can be arbitrarily small.

3.1 Topological charge

The spectrum of a system with a continuous translational symmetry is parameterized by a wave vector that “resides” on a plane. As a result, continuous systems without intrinsic periodicity generally lack a well-defined topological classification [9]. However, it has been shown that the response of such systems can always be regularized by introducing a high-spatial frequency cutoff, which suppresses nonreciprocal effects at short wavelengths (k → ∞) [9]. Therefore, since the Berry dipole material is nonreciprocal, its regularized response can be associated with nontrivial topological phases.

The simplest way to account for material dispersion and non-Hermitian effects is to formulate the electromagnetic problem in a manner that explicitly incorporates the relevant physical degrees of freedom responsible for the dispersive response. In Appendix A, this is done for our system using a phenomenological transport equation that effectively describes how the material’s free electrons are influenced by the external electric field and by the anomalous velocity term arising from the geometry of the electronic bands via the Berry dipole. This approach reduces the spectral problem to the canonical form L k ⋅ Q = ω c Q , where Q is a state vector defined in Appendix A and L _k is the matrix operator in Eq. (A5).

The topological charge of a band gap can be expressed in terms of the system Green’s function ( G k = i L k − ω c 1 − 1 ) through an integral in the complex plane, taken over a line within the band gap that runs from ω _gap − i∞ to ω _gap + i∞ [26], [27], [28], [29], [30]. Here, ω _gap represents a frequency within the gap. The relevant gap in our system is shaded in pink in Figure 2b and c, and takes the form ω _gap,L < ω′ < ω _gap,H with ω gap,L = ω p 2 − Γ / 2 2 . Another low-frequency gap is visible in the figure, but this gap is closed in the presence of a finite wave vector cutoff and, therefore, holds no topological significance (not shown).

Using the Green’s function approach, we have determined that the high-frequency band gap is topological, characterized by the gap Chern number:

Thus, the sign of the topological charge is strictly tied to the sign of the product of the Berry dipole and to the orientation of the electric bias. By reversing the orientation of the electric bias, a topological phase transition can be triggered, causing the Chern number to reverse its sign.

3.2 Topological edge-states

In Hermitian systems, the “bulk-edge correspondence” establishes a precise link between the gap Chern numbers of two topological materials and the net number of unidirectional edge states supported by a material interface [10], [11], [12], [13]. In non-Hermitian systems, most notably in systems with gain, it has been shown that this correspondence may break down due to a phenomenon known as the “non-Hermitian skin effect” [19], [20], [21], [22]. This effect occurs when the bulk spectrum undergoes a dramatic shift as the system transitions from periodic boundary conditions to “opaque” boundaries, commonly referred to as “open boundaries” in electronic systems. In systems exhibiting the non-Hermitian skin effect, bulk states tend to be exponentially localized at the boundary. To our knowledge, the non-Hermitian skin effect only occurs in systems with gain, with the bulk spectrum of the relevant operator spanning both the lower and upper halves of the frequency plane [23], [24]. In particular, for systems with a spectrum confined to the lower-half frequency plane, the bulk-edge correspondence may remain valid.

To investigate the correlation between the gap Chern number and the emergence of topological edge states, we characterized the dispersion of boundary modes when the chiral-gain medium (region y > 0) is paired with a metal with permittivity ɛ _m (region y < 0), as shown in Figure 1b. The dispersion equation of the TM-polarized edge modes is described by [9]:

Here, k _x is the propagation constant of the edge state, ɛ _D, ɛ _EO are the diagonal and anti-diagonal elements of the permittivity tensor of the Berry dipole material (Eq. (3)), γ = k x 2 − ε ef ω c 2 and γ m = k x 2 − ε m ω c 2 are the attenuation constants of the Berry dipole material and metal, respectively, along the y-axis, and ε ef = ε D 2 − ε EO 2 / ε D .

For simplicity, in this section, we consider the metal as a perfect electric conductor (PEC), corresponding to ɛ _m = −∞. In this case, the edge-state dispersion equation simplifies to ɛ _D γ = ɛ _EO k _x . Squaring both sides, some algebra reveals that the solutions of this equation must also satisfy ε D ω c 2 = k x 2 . However, the reverse is not true; some solutions of the simplified equation do not satisfy the original dispersion. Interestingly, since ɛ _D is independent of the bias in our problem, we find that the edge-state dispersion is also independent of ω _c!

Figure 4a represents a parametric plot of the complex eigenfrequencies ω′ + iω″ of the edge-states as a function of k _x real-valued. This dispersion models an edge waveguide that is closed on itself in the form of a loop. When ω _c > 0, the solutions with Re ω > 0 are linked to k _x > 0, indicating that the edge modes are unidirectional and propagate along the +x-axis. The edge state has a finite lifetime ( Im ω < 0 ) because of the material absorption. As shown in Figure 4a, the edge states dispersion is gapless (green curve) and connects the two bulk bands, in accordance with the bulk-edge correspondence. Moreover, the propagation direction for the edge state (along +x) is also consistent with the sign of the gap Chern number (C _gap = −1), as predicted by the bulk edge correspondence for Hermitian systems [13]. Note that, similar to the bulk case, there exists a branch of static-like edge-state solutions confined to the imaginary axis.

Figure 4:

Complex dispersion of the edge-states (green and cyan lines) at the interface between the chiral-gain medium and a trivial medium (metal). The chiral gain medium is characterized by Γ = 0.5ω _p . The black triangles mark the short-wavelength solutions (k _x → ±∞) obtained with the quasi-static approximation. The black arrows indicate the direction of increasing k x along each branch of the edge-state dispersion curve. The purple lines represent the bulk modes. (a) Interface between a chiral gain medium with ω _c = 0.5ω _p and a PEC. (bi) Interface between a chiral gain medium with ω _c = 0.5ω _p and a low-loss metal with plasma frequency ω _m = 8ω _p. (bii) Close-up of the dispersion of the edge-states depicted in panel (bi). In all the panels, for ω′ > 0, the green lines are associated with edge states with a positive k _x , whereas the cyan lines are associated with edge states with a negative k _x .

A few observations are in order. First, for simplicity, we neglect the effect of the high-frequency spatial cutoff in the calculation of the edge states. This could be accounted for using an approach similar to that described in Ref. [12], which involves additional boundary conditions at the PEC interface, specifically the vanishing of the electric current density j defined in Appendix A. The second observation pertains to the limit ω _c → 0. As mentioned, the edge-state dispersion is independent of bias strength but depends on the sign of ω _c. For positive (negative) ω _c, allowed edge states with Re ω > 0 exhibit k _x > 0 (k _x < 0), implying a discontinuous transition at ω _c = 0. This behavior arises because the linearized model of the material response assumes a static bias much stronger than the dynamical signal [48], making the model ill-defined in the limit ω _c → 0. A final observation is that, since the edge states for a PEC interface are insensitive to the bias strength, it follows that even if the bulk material is unstable, the dispersion of the topological edge state always lies in the lower-half frequency plane. Thus, for this configuration, it is impossible to exploit the gain properties of the Berry dipole material to induce amplifying unidirectional edge states.

4.1 Gain momentum locking

It is well-known that surface plasmons in metals possess an intrinsic spin determined by their direction of propagation [56], [58], [59]. Specifically, the direction of the spin angular momentum σ of the surface plasmons is determined by the cross product of the momentum (k) and the attenuation direction (γ). This property, known as spin-momentum locking [56], [58], [59], implies that plasmons propagating in the counter-clockwise direction with respect to the z-axis have their spin angular momentum pointing along the positive z axis, while those propagating clockwise direction have their spin angular momentum oriented along the –z axis.

Interestingly, as discussed in Section 2.2, in a chiral gain medium, one spin state is associated with material dissipation (Figure 5bi), whereas the opposite spin state leads to material gain (Figure 5bii). It was recently shown in Ref. [57] that the interplay between spin-momentum locking and chiral gain results in gain-momentum locking: plasmons with a propagation direction such that the spin angular momentum is parallel to the gain vector Ω _ω are amplified within the medium (Figure 5bi), while those with opposite propagation direction experience absorption (Figure 5bii). The analysis in Ref. [57] relies on a simplified, nondispersive permittivity model for the chiral-gain medium. Building on these findings, we now explore how, in more realistic dispersive systems, the interplay between spin and gain can be leveraged to engineer edge states with amplification.

It is important to note that for ω _c > 0, the gap Chern number is negative, meaning that if there is a single topological state, the bulk-edge correspondence dictates it must propagate in the counterclockwise direction within the cavity geometry [13]. However, from Eq. (9), we see that for ω _c > 0, the gain vector is directed along the –z axis, i.e., anti-parallel to the plasmon spin angular momentum. Thus, in this system, the topological state is necessarily associated with dissipation. This offers a new perspective on the findings in Section 3.2 and further explains why the single topological state supported by the chiral-gain medium and a PEC wall is unaffected by optical gain.

4.2 Unidirectional edge-states with optical gain

From the previous subsection, it is clear that for the considered dispersion model of the Berry dipole material, optical gain can only be observed in a spectral region outside the topological bandgap. Additionally, the cavity walls cannot be perfectly conducting. Taking this into account, we now assume the cavity walls follow a dispersive Drude model described by the permittivity ε m ω = 1 − ω m 2 / ω 2 where ω _m is the plasma frequency of the metal. For simplicity, we assume that the dissipation in the cavity walls is negligible in this analysis.

Furthermore, similar to previous examples, we assume that the Berry dipole material is characterized by Γ = 0.5ω _p, and we suppose that the bias strength is slightly below the instability threshold ω _c,th ≈ 0.5ω _p to maximize the gain effect. It should be noted that for the case of p-doped tellurium this bias strength is about 100 times larger than what has been considered in experiments [48]. This problem can be alleviated by using n-doped tellurium or other engineered material with a much stronger Berry dipole [48]. The metal is characterized by ω _m = 8 ω _p.

The calculated edge-state dispersion (computed from Eq. (11)) is depicted in Figure 4bi (green and cyan lines). It corresponds to a parametric plot of the complex eigenfrequencies ω′ + iω″ as a function of k _x real-valued. The purple lines in Figure 4bi represent the bulk dispersion, which is totally confined to the lower-half frequency plane. As seen, now there are two branches of edge states with ω′ > 0. The black triangles mark the short-wavelength solutions (k _x → ±∞), which are obtained using a quasi-static formalism described in Appendix B.

Similar, to the example of Section 3.2, one of the edge-state branches (colored in green) represents the gapless topological modes. In agreement with previous considerations, it is fully contained in the lower-half frequency plane. Remarkably, in the frequency window 3ω _p ≤ ω′ ≤ 6ω _p, there is an additional branch (colored in cyan) that is partially contained in the upper-half frequency plane (ω″ > 0) resulting in optical gain (see a zoom in Figure 4bii). In agreement with the gain-momentum locking principle, this branch of edge states is associated with plasmons that propagate in the clockwise direction (–x axis for the interface y = 0 in Figure 1b). In the frequency window, 3ω _p ≤ ω′ ≤ 6ω _p, one of the eigenvalues of ε ̄ ′ ′ is negative (see Figure 3d), confirming that the Berry dipole material has a chiral gain response.

It should be noted that amplifying edge states can only be excited when the Berry dipole material is paired with a material with negative permittivity, as considered here. Indeed, as discussed in Section 2.1, the Berry dipole material exhibits gain only when Re ε D > 0 , and, as is well known, the excitation of edge states requires that Re ε m Re ε D < 0 .

Remarkably, the orbital angular momentum of the lasing mode is intrinsically linked to the orientation of the bias electric field. This makes the proposed system useful for generating structured light with inherent orbital angular momentum. Moreover, reversing the direction of the bias electric field E ₀, causes the lasing mode to propagate in the opposite direction. Lasing relying on bulk modes in Berry dipole materials was previously discussed in Refs. [60], [61].

To further illustrate these ideas, we calculated the fields associated with the edge states for the interface y = 0 (Figure 1b). The magnetic field complex amplitude is given by:

Figure 6b represents a time snapshot of the field profile of the two edge states with ω/ω _p = 4, for the same material parameters as in Figure 4b. Here, we assume open boundary conditions so that ω is taken as real-valued, whereas the propagation constant of the edge state k _x is complex-valued. Consistent with the chiral gain response of the material and with the gain momentum locking, plasmons that propagate along the –x direction have a spin that matches the eigenpolarization that activates the gain resulting in amplification (Figure 6bi). On the other hand, plasmons that propagate along the +x direction activate dissipation in the material resulting in absorption (Figure 6bii).

Figure 6:

Density plot of the magnetic field associated with the boundary modes for the interface y = 0. The black arrows indicate the direction of propagation of the edge state. (ai) Plasmon propagating in the −x direction resulting in amplification. (aii) Plasmon propagating in the +x direction resulting in attenuation.