Engineering non-Hermitian skin effect with band topology in ultracold gases

Abstract

Non-Hermitian skin effect(NHSE) describes a unique non-Hermitian phenomenon that all eigen-modes are localized near the boundary, and has profound impact on a wide range of bulk properties. In particular, topological systems with NHSE have stimulated extensive research interests recently, given the fresh theoretical and experimental challenges therein. Here we propose a readily implementable scheme for achieving NHSE with band topology in ultracold gases. Specifically, the scheme realizes the one-dimensional optical Raman lattice with two types of spin-orbit coupling (SOC) and an additional laser-induced dissipation. By tuning the dissipation and the SOC strengths, NHSE and band topology can be individually controlled such that they can coexist in a considerable parameter regime. To identify the topological phase in the presence of NHSE, we have restored the bulk-boundary correspondence by invoking the non-Bloch band theory, and discussed the dynamic signals for detection. Our work serves as a guideline for engineering topological lattices with NHSE in the highly tunable environment of cold atoms, paving the way for future studies of exotic non-Hermitian physics in a genuine quantum many-body setting.

Introduction

Open quantum systems undergoing particle or energy loss can be effectively described by non-Hermitian Hamiltonians. They exhibit intriguing non-Hermitian phenomena that are absent in their Hermitian counterparts and have thus attracted significant attention in recent years^1,2. An outstanding example here is the non-Hermitian skin effect (NHSE)^{3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20}, under which all bulk eigenstates are localized near the boundary. While NHSE is topologically protected by the winding of eigen-spectrum in the complex energy plane^12,13, the spectrum itself is sensitive to the actual boundary condition. For instance, both the eigen-spectrum and eigen-wavefunction can be dramatically different under an open boundary condition (OBC) from those under a periodic boundary condition (PBC). A remarkable consequence is the failure of conventional bulk-boundary correspondence in topological systems with NHSE, whose restoration calls for the so-called non-Bloch band theory by employing a generalized Brillouin zone (GBZ)^3,4,5,6. Apart from the fundamental impact on band topology, NHSE can strongly influence many other bulk properties such as the dynamics^{9,16,21,22,23}, the parity-time symmetry^24,25, and localization^26,27.

To date, NHSE has been observed in various non-Hermitian one-dimensional (1D) topological systems, including photonics^28,29,30, topoelectrical circuits³¹, and metamaterials³², wherein the non-Bloch bulk-boundary correspondence has also been confirmed^28,29,31. In these studies, NHSE is predominantly achieved through nonreciprocal hopping, by simulating either the Hatano-Nelson model³³ or the nonreciprocal Su-Schrieffer-Heeger model³. At the moment, the study of NHSE deserves a substantial extension to a broader context. On the one hand, the appearance of NHSE is not limited to these models. For instance, a simple spin rotation in momentum space can directly convert the nonreciprocal hopping to on-site dissipation^3,34, under which NHSE persists^7,9,14,15,16. On the other hand, given the existing experiments are either classical or on the level of single photons, it is desirable to engineer NHSE in a quantum many-body setting, which would offer exciting opportunities for investigating the interplay of NHSEs with many-body statistics and interactions.

Ultracold atomic gases, with highly controllable parameters, are an ideal candidate for the task. In this platform, through the photon-mediated Raman coupling technique, both the 1D and 2D spin-orbit couplings (SOC) have been realized^{35,36,37,38,39,40,41,42,43}, culminating in the successful generation of topological bands in optical lattices^{42,43,44,45,46}. Meanwhile, laser-induced atom loss has enabled the experimental realization of parity-time symmetry in ultracold atoms^47,48,49, and a very recent experiment manages to incorporate the SOC with laser-induced loss in a single setup⁴⁹.

In light of these achievements, we propose to engineer NHSE with band topology in ultracold atoms by utilizing the Raman-assisted SOCs and laser-induced atom loss. Specifically, we consider a one-dimensional optical Raman lattice with two distinct types of SOCs, where the nontrivial band topology is facilitated by one type of SOC, and the NHSE originates from the other, along with the laser-induced loss. While both types of SOC are indispensable in inducing the NHSE with nontrivial band topology, they are not a trivial combination but exhibit a strong interplay effect in this process. We have mapped out the topological phase diagram, tabulated the parameter regimes for NHSE and band topology, and proposed the dynamical detection scheme. As all aspects of our proposal are readily accessible, our work represents a significant step toward the observation of NHSEs and the associated exotic phenomena in a genuine quantum many-body setting.

Results

Model

As illustrated in Fig. 1, we consider a two-component(↑,↓) atomic gas in a 1D optical lattice (along x), described by the Hamiltonian H = ∫dxH(x), with

$$H(x) = \frac{{p}_{x}^{2}}{2m}-{V}_{0}\cos (2{k}_{0}x)+{M}_{0}\sin {k}_{0}x({e}^{i{\phi }_{0}}{\sigma }_{+}+H.c.)\\ +{M}_{r}({e}^{i2{k}_{r}x}{\sigma }_{+}+h.c.)+i\gamma {\sigma }_{z}.$$

(1)

Here σ_± = σ_x ± iσ_y, with σ_α(α = x, y, z) the Pauli matrices, and $\frac{{p}_{x}^{2}}{2m}$ is the kinetic term.

Fig. 1: Schematics of the experimental setup in generating two types of spin-orbit couplings (SOCs).

a Two sets of Raman lasers couple the ground spin states via electronically excited states (dashed). b Laser configuration and optical-path diagram. The green and orange lines show the propagation directions of the two sets of lasers in a, and the black arrows indicate their polarizations. L_1x, L_1z, and L₂ are, respectively, the optical paths from the sample to mirror M₁, from the beam splitting to mirror M₂, and from the sample to M₄ (and M₅). The optical lattice potential and the Ω₀-SOC are generated by electric fields (E_1x, E_1z), and the Ω_r-SOC is created by electric fields (${{{{{{{{\bf{E}}}}}}}}}_{2x},{{{{{{{{\bf{E}}}}}}}}}_{2x}^{\prime}$). c The Ω₀- and Ω_r-SOCs, respectively, generate the nearest-neighbor and the on-site spin flip with tunable strengths ${\Omega }_{0}{e}^{i{\phi }_{0}}{(-1)}^{j}$ and ${\Omega }_{r}{e}^{i{\phi }_{r}j}$ (j is the site index).

The optical Raman lattice^42,43,46, characterized by V₀ and M₀, is generated by two Raman lasers: a standing wave propagating along x with the electric-field vector ${{{{{{{{\bf{E}}}}}}}}}_{1x}={{{{{{{{\bf{e}}}}}}}}}_{z}2{E}_{1x}{e}^{i{\phi }_{1+}}\cos ({k}_{0}x+{\phi }_{1-})$, where ${\phi }_{1\pm }=({\phi }_{1x}\pm {\phi }_{1x}^{\prime})/2$, and ${\phi }_{1x}\,({\phi }_{1x}^{\prime})$ is the phase of incident (reflected) light; and a plane wave propagating along z with ${{{{{{{{\bf{E}}}}}}}}}_{1z}={{{{{{{{\bf{e}}}}}}}}}_{x}{E}_{1z}{e}^{i{k}_{0}z+{\phi }_{1z}}$. As shown in Fig. 1b, E_1x and E_1z come from the same laser source (laser 1) through a beam splitter, enabling easy manipulation of various relative phases. For instance, the phase ϕ₁₋( = − k₀L_1x) is adjustable through the optical-path L_1x from the sample (gray dot) to mirror M₁. By taking ϕ₁₋ = − π/2 and z = 0, the field E_1x generates a lattice potential with spacing a = π/k₀ and lattice sites at x_i = ia. E_1x, E_1z combine to form the SOC that couples different spins with amplitude ${M}_{0}\sin ({k}_{0}x)$ and phase ϕ₀ = ϕ₁₊ − ϕ_1z. The phase ϕ₀ = k₀(L_1x − 2L_1z) is tunable through L_1x or L_1z, with L_1z the perpendicular distance (along $\hat{z}$ between the mirrors M₂, M₃ and the sample).

An additional SOC, characterized by M_r and known as the equal Rashba and Dresslehaus coupling^{35,36,37,38,39,44,45}, is created by two plane-wave Raman lasers(E_2x,${{{{{{{{\bf{E}}}}}}}}}_{2x}^{\prime}$) with opposite wave-vectors along x and equal phase. As shown in Fig. 1b, E_2x and ${{{{{{{{\bf{E}}}}}}}}}_{2x}^{\prime}$ are both from the laser source 2 (with wave-vector k₀) and intersect at the sample following reflections by two mirrors M₄, M₅. By adjusting the angle θ between their propagation directions and $\hat{x}$, the recoil momentum in (1) is tunable as ${k}_{r}={k}_{0}\cos \theta$. Their relative phase can be tuned to zero, as is the case with (1), by adjusting the optical-path L between the mirrors and the sample such that $2{k}_{0}{L}_{2}(1+\sin \theta )=2n\pi$, with $n\in {\mathbb{Z}}$.

To ensure the atomic transitions as illustrated in Fig. 1a and the zero detuning between different spins, it is required that the laser-frequency difference between {E_1x, E_1z}, and that between $\{{{{{{{{{\bf{E}}}}}}}}}_{2x},\,{{{{{{{{\bf{E}}}}}}}}}_{2x}^{\prime}\}$, exactly match the Zeeman splitting between ↑ and ↓. This can be achieved through the acousto-optical modulater (AOM), which has been widely used for two-photon Raman processes in cold atoms experiments^{35,37,40,41,42,46}. Note that we have not shown AOM explicitly in Fig. 1.

A laser-induced loss term, characterized by γ, is generated by coupling spin-↓ atom to an excited state that is subsequently lost from the system due to spontaneous emission^47,48,49. The conditional dynamics of the system under post-selection are characterized by the non-Hermitian Hamiltonian (1), after dropping the global loss term ( ~ iγ).

The tight-binding model corresponding to Eq.(1) is

$$H = -t\mathop{\sum}\limits_{j}\left({c}_{j\uparrow }^{{{{\dagger}}} }{c}_{j+1\uparrow }+{c}_{j\downarrow }^{{{{\dagger}}} }{c}_{j+1\downarrow }+h.c.\right)\\ +{\Omega }_{0}\mathop{\sum}\limits_{j}\left[{e}^{i{\phi }_{0}}{(-1)}^{j}\left({c}_{j\uparrow }^{{{{\dagger}}} }{c}_{j+1\downarrow }-{c}_{j\uparrow }^{{{{\dagger}}} }{c}_{j-1\downarrow }\right)+h.c.\right]\\ +{\Omega }_{r}\mathop{\sum}\limits_{j}\left({e}^{i{\phi }_{r}j}{c}_{j\uparrow }^{{{{\dagger}}} }{c}_{j\downarrow }+h.c.\right)+i\gamma \mathop{\sum}\limits_{j}\left({c}_{j\uparrow }^{{{{\dagger}}} }{c}_{j\uparrow }-{c}_{j\downarrow }^{{{{\dagger}}} }{c}_{j\downarrow }\right).$$

(2)

Here the two SOCs respectively provide the nearest-neighbor and the on-site spin flip with amplitudes Ω₀ and Ω_r. We henceforth denote them as the Ω₀- and Ω_r-SOC, respectively, along with the phase parameters ϕ₀( = ϕ₁₊ − ϕ_1z) and ϕ_r( = 2πk_r/k₀). Since all the parameters in (2) are highly tunable, in this work, we fix the hopping rate t as the energy unit (t = 1) and take ϕ₀, ϕ_r ∈ [0, 2π).

Our scheme is applicable to a wide range of alkali and alkali-earth(-like) atoms. A promising candidate is ¹⁷³Yb, where both the optical Raman lattice with band topology⁴⁶ and non-Hermitian SOC⁴⁹ have been realized using the Raman-induced ¹S₀ ↔ ³P₁ transitions. A detailed Raman-transition scheme corresponding to Fig. 1a for ¹⁷³Yb can be found in [Supplementary Note 1].

NHSE and band topology

To provide insight and highlight the individual role of two SOCs, we consider the following three cases:

Case-I: Ω₀ = 0, Ω_r ≠ 0

This is the lattice version of the continuum gas with non-Hermitian SOC, as implemented recently in ref. ⁴⁹. Here, following the gauge transformation ${c}_{j\downarrow }\to {c}_{j\downarrow }{e}^{-i{\phi }_{r}j}$, we obtain the Bloch Hamiltonian $H(k)=-2t\cos ({\phi }_{r}/2)\cos \tilde{k}+{\Omega }_{r}{\sigma }_{x}+(i\gamma -2t\sin ({\phi }_{r}/2)\sin \tilde{k}){\sigma }_{z}$ (with $\tilde{k}=k+{\phi }_{r}/2$) and the eigenenergy

$${E}_{k\pm }^{(I)} = -2t\cos ({\phi }_{r}/2)\cos \tilde{k}\\ \pm \sqrt{{\Omega }_{r}^{2}+{(i\gamma -2t\sin ({\phi }_{r}/2)\sin \tilde{k})}^{2}}.$$

(3)

The eigen-spectrum supports two exceptional points at Ω_r = γ, when $\tilde{k}=0,\pi$ or ϕ_r = 0.

Importantly, the system hosts NHSE for ϕ_r ≠ 0, π, as clearly indicated by the closed-loop topology^12,13 of (3) in the complex plane. The presence of NHSE leads to distinct spectra under PBC and OBC, and localized bulk eigen-modes near the OBC boundary (see Fig. 2a (ii), b (ii)). Nevertheless, this case is trivial in band topology, since H(k) does not show any spin-winding as k traverses the Brillouin zone. As a result, the OBC spectrum does not feature any in-gap topological modes (see Fig. 2a (i), (ii)).

Fig. 2: Non-Hermitian skin effect and band topology for cases I and II.

Eigen-spectrum (a (i–iii)) and spatial profile of eigen-modes (b (i–iii)) for case-I and -II when only one spin-orbit coupling (SOC) is present. Here we take the dissipation strength γ = 0.2, and the other parameters are (Ω₀, ϕ₀, Ω_r, ϕ_r) = (0, − , 0.3, π) for a (i), b (i); (0, − , 0.3, π/2) for ((a)ii,(b)ii); (0.5, π/2, 0, − ) for a (iii), b (iii). The non-Hermitian skin effect shows up only in a (ii), b (ii). a (iii), b (iii) are topological with two in-gap zero modes localized at both boundaries. Here the energy unit is taken as hopping t.

Case-II: Ω₀ ≠ 0, Ω_r = 0

This is the case with only optical Raman lattice and dissipation. Following the transformation c_j↓ → (−1)^jc_j↓, the Bloch Hamiltonian $H(k)=2{\Omega }_{0}\sin k(\sin {\phi }_{0}{\sigma }_{x}+\cos {\phi }_{0}{\sigma }_{y})+(i\gamma -2t\cos k){\sigma }_{z}$, and the eigen-spectrum is

$${E}_{k\pm }^{(II)}=\pm \sqrt{{\left(2{\Omega }_{0}\sin k\right)}^{2}+{\left(i\gamma -2t\cos k\right)}^{2}}.$$

(4)

Clearly, there is no NHSE—the spectrum (4) exhibits no loop structures in the complex plane. However, H(k) possesses nontrivial band topology, as further confirmed in Fig. 2a (iii), b (iii) by the appearance of in-gap zero modes under OBC and their localized wave functions near boundaries. The topological transition occurs at γ = 2Ω₀ for all ϕ₀, when the band gap closes at k = π/2, 3π/2. We note that the topological phase of a similar model under a real Zeeman field (iγ → Γ_z) and ϕ₀ = 0 has been studied in ref. ⁵⁰, where the topological transition occurs at Γ_z = 2t.

One can see that the Ω_r-SOC and the Ω₀-SOC can, respectively, give rise to NHSE and band topology, as they respectively lead to spectral and wavefunction windings. In order to achieve both in a single setting, one needs to incorporate all essential gradients to satisfy both winding conditions. A natural contender is by combining both types of SOCs, as well as the on-site loss.

Case-III: Ω₀ ≠ 0, Ω_r ≠ 0

When both types of SOCs are switched on, an analytical form of the eigen-spectrum is generally unavailable. An exception is when ϕ_r = π, under which the two SOCs are commensurate, and k is still a good quantum number. The Bloch Hamiltonian is $H(k)=({\Omega }_{r}+2{\Omega }_{0}\sin {\phi }_{0}\sin k){\sigma }_{x}+2{\Omega }_{0}\cos {\phi }_{0}\sin k{\sigma }_{y}+(i\gamma -2t\cos k){\sigma }_{z}$, with the eigen-spectrum

$${E}_{k\pm }^{({{{{{{{\rm{III}}}}}}}},{\phi }_{{{{{{{{\rm{r}}}}}}}}} = \pi )} = \pm \left[\left({\Omega }_{r}^{2}+{\left(2{\Omega }_{0}\sin k\right)}^{2}+4{\Omega }_{r}{\Omega }_{0}\sin {\phi }_{0}\sin k\right.\right.\\ {\left.+{\left(i\gamma -2t\cos k\right)}^{2}\right]}^{1/2}.$$

(5)

We see immediately that once ϕ₀ ≠ 0, π, the spectrum (5) would form a loop in the complex plane, signifying the presence of NHSE. Moreover, H(k) in this case keeps a similar spin-winding pattern as in case-II, and thus the system acquires a band topology. In particular, when ϕ₀ = π/2, 3π/2 the system has the chiral symmetry: σ_yH(k)σ_y = − H(k), which protects the degenerate topological zero modes. In Fig. 3, we have numerically verified the coexistent skin and topological properties for ϕ_r = π and ϕ₀ = π/2, from both the different eigen-spectra between PBC and OBC (Fig. 3a), and the localized bulk state and in-gap zero modes (Fig. 3b).

Fig. 3: Coexistence of non-Hermitian skin effect and band topology for case-III with ϕ_r = π, ϕ₀ = π/2, and Ω₀ = 0.5, Ω_r = 0.3, γ = 0.2.

a Energy spectra under periodic boundary condition and open boundary condition in the complex plane. b Spatial wave functions for the two topological edge modes and two randomly chosen bulk states. Here the energy unit is taken as hopping t.

The band topology can be conveniently tuned by γ and Ω₀, Ω_r. In Fig. 4a (i), we take a specific set of Ω₀ and Ω_r, and show that by increasing γ to a critical γ_c, the in-gap zero modes merge into the bulk and the gap closes and reopens across this critical point. This signifies a topological transition into the trivial phase. Remarkably, γ_c is different from the gap-closing point of the eigen-spectrum (5) under PBC, where γ_c,PBC = ∣Ω_r ± 2Ω₀∣. This is exactly the breakdown of conventional bulk-boundary correspondence under NHSE.

Fig. 4: Topological phase transition for case-III with ϕ_r = π, ϕ₀ = π/2.

a (i, ii) The amplitude of spectrum ∣E∣ under open boundary condition (blue) and the winding number W obtained from ed Brillouin zone as functions of γ. Here we take Ω₀ = 0.5, Ω_r = 0.3. The topological transition occurs at γ_c ~ 1, which differs from the periodic boundary condition predictions γ_c,PBC = 0.7, 1.3 (gray arrows). The periodic boundary condition eigen-spectrum is shown in gray for comparison. b Contour plot of γ_c in (Ω₀, Ω_r) plane (here the color bar represents the value of γ_c). The energy unit is taken as hopping t.

To restore the bulk-boundary correspondence, we adopt the non-Bloch band theory^3,4,5,6 and compute the winding number W in the GBZ. As shown in Fig. 4a (ii), W can well predict the topological transition under OBC: the in-gap zero modes emerge where W = 1 (γ < γ_c) and vanish where W = 0 (γ > γ_c). In fact, γ_c can be obtained analytically through the gap-closing condition of the GBZ spectrum[Methods]. The resulting topological phase diagram is given in Fig. 4b, where γ_c is plotted as functions of Ω₀ and Ω_r for the case of ϕ₀ = π/2. The ϕ₀ = 3π/2 case is found to share the same diagram due to symmetries[Methods].

When (ϕ_r, ϕ₀) deviate from (π, π/2) and (π, 3π/2), the chiral symmetry is broken, and the topological modes would split and gradually merge into the bulk. In comparison, NHSE is much more robust, which can persist for all (ϕ_r, ϕ₀) except for ϕ_r = 0 and two discrete points (π, π) and (π, 0)[Supplementary Note 2]. In Table 1, we summarize the conditions for band topology and NHSE for cases I–III. The main message is that both SOCs are indispensable in achieving NHSE and topology simultaneously. Their combination shows an intriguing interplay effect, instead of being a trivial superposition. For instance, the application of Ω_r-SOC in case-III changes the topology condition as compared to case-II, and the Ω₀- SOC changes the skin condition as compared to case-I. In particular, for the latter case, the presence of Ω₀-SOC further broadens the parameter region of observing skin modes, signifying a positive effect of band topology in enhancing NHSE.

Table 1 Conditions for achieving topological phase and non-Hermitian skin effect (NHSE) for various cases.

Dynamic detection

To detect the topological phase with NHSE, we propose an edge-to-edge transport measurement. In the Hermitian case, the topological edge modes play the dominant role in such transport ⁵¹. However, in the presence of NHSE, the transport is expected to be significantly modified since all bulk modes also localize near the edge. To examine such an effect, we compare two topological systems in our setup, one is Hermitian at γ = 0, and the other is with NHSE at finite γ. We study the probability of particle occupation at the β-edge of the system at time τ, when the initial state starts from the α-edge (α, β = L or R)

$${P}_{\alpha \beta }(\tau )={\left|\left\langle \beta \right|{e}^{-iH\tau }\left|\alpha \right\rangle \right|}^{2}.$$

(6)

To eliminate the difference caused by spin, we take the initial state as the equal population of ↑ and ↓ and show its dynamics in Fig. 5. In the Hermitian case (Fig. 5a), the left-to-right (P_LR) and right-to-left (P_RL) transports are identical. As a manifestation of the topological edge states, the oscillation frequency of P_LR (or P_RL), as given by the energy gap between the two edge modes in a finite-size system, is found to decay exponentially with increasing system size (inset of Fig. 5a). In the presence of NHSE (Fig. 5b), the transport properties are dramatically different. Due to the localization of skin modes at the left boundary, the transport shows a strong directional preference towards the left side, namely. In this case, the topological edge modes play little role in affecting the dynamics (inset of Fig. 5b). These features distinguish the topological phases with and without NHSE.

Fig. 5: Topological transport with and without non-Hermitian skin effect.

Edge-to-edge transport properties for the topological system without (a) and with (b) non-Hermitian skin effect. We take Ω₀ = 0.5, Ω_r = 0.3, and γ = 0 (a), 0.2 (b). In the main plots, the system size L = 8. The inset of b shows the contribution from the topological edge states. Here the energy unit is hopping t, and the time unit is ℏ/t.

Optical lattices with sharp boundaries can be implemented using box-trap potentials^{52,53,54,55,56}, where the spatial extent of an edge is determined by the optical wavelength ~1 μm (much smaller than the typical trap length ~10–100 μm). Such a small imperfection does not visibly change the dynamics in Fig. 5 [Supplementary Note 3]. Even sharper edges (~10 nm) can be created via the dark state in atomic Λ-system^57,58,59. Alternatively, without edges, the NHSE can also manifest itself in bulk dynamics^16,21,22. Indeed we have confirmed that NHSE can lead to visible directional bulk transport under typical harmonic trapping potentials[Supplementary Note 3].

For the detection of the non-Bloch band topology, one may further resort to quench dynamics or measurement of the biorthogonal chiral displacement^60,61. Alternatively, topological edge states may be probed through a time-integrated state tomography²⁸.

Discussion

We have proposed a realistic scheme for utilizing the SOCs and the spin-dependent dissipation in ultracold atoms to engineer NHSE with band topology. We emphasize that both SOCs are indispensable in the scheme. Their mutual interference, along with their interplay with the on-site dissipation, determines the ultimate parameter regime for skin modes with nontrivial band topology (Table 1).

For future studies, an intriguing possibility would be tuning ϕ_r away from π such that the two SOCs become incommensurate. The competition between quasiperiodicity and NHSE would potentially lead to unique localization features^26,27. Further generalization of our scheme to higher dimensions would offer the opportunity for achieving Weyl exceptional rings⁶², high-order skin effect, and band topology featuring corner or hinge modes^63,64,65. Moreover, the implementation of NHSE in ultracold atoms paves the way for exploring collective phenomena therein due to interatomic interactions, which are easily tunable through Feshbach resonances. Our proposal, therefore, ushers in a wide variety of possibilities for the quantum simulation of non-Hermitian physics.

Methods

Derivation of the tight-binding model

To derive the tight-binding model, we expand the field operator ψ_σ(x) = ∑_iω_n=0(x − x_i)c_iσ, where ω_n=0(x) is the lowest-band Wannier function, and i is the index of lattice sites. In this way, the second-quantized single-particle Hamiltonian can be reduced to the tight-banding model, with the following parameters:

(1) the nearest-neighbor hopping term

$$t=-\int dx{\omega }_{0}(x)\left(\frac{{p}_{x}^{2}}{2m}-{V}_{0}\cos (2{k}_{0}x)\right){\omega }_{0}(x-a).$$

(7)

(2) the on-site spin-flip terms

$${t}_{\uparrow \downarrow }^{j}=\int dx{\omega }_{0}^{* }(x-{x}_{j}){M}_{r}{e}^{i2{k}_{r}x}{\omega }_{0}(x-{x}_{j})\equiv {e}^{i{\phi }_{r}j}{\Omega }_{r},$$

(8)

$${t}_{\downarrow \uparrow }^{j}=\int dx{\omega }_{0}^{* }(x-{x}_{j}){M}_{r}{e}^{-i2{k}_{r}x}{\omega }_{0}(x-{x}_{j})\equiv {e}^{-i{\phi }_{r}j}{\Omega }_{r},$$

(9)

where the amplitude ${\Omega }_{r}={M}_{r}\int dx{\omega }_{0}^{2}(x){e}^{i2{k}_{r}x}$, and the corresponding phase ϕ_r = 2k_ra = 2πk_r/k₀. Note that here we pin down the coordinate of the j-site atom as x_j = ja, with a = π/k₀ as the lattice spacing.

(3) the nearest-neighbor spin-flip terms

$${t}_{\uparrow \downarrow }^{j,j+1} = \int dx{\omega }_{0}^{* }(x-{x}_{j})({M}_{0}\sin {k}_{0}x{e}^{i{\phi }_{0}}){\omega }_{0}(x-{x}_{j+1})\\ \equiv \, {e}^{i{\phi }_{0}}{\left(-1\right)}^{j}{\Omega }_{0},$$

(10)

$${t}_{\uparrow \downarrow }^{j,j-1} = \int dx{\omega }_{0}^{* }(x-{x}_{j})({M}_{0}\sin {k}_{0}x{e}^{i{\phi }_{0}}){\omega }_{0}(x-{x}_{j-1})\\ \equiv -{e}^{i{\phi }_{0}}{\left(-1\right)}^{j}{\Omega }_{0},$$

(11)

where the amplitude ${\Omega }_{0}={M}_{0}\int dx{\omega }_{0}(x)\sin {k}_{0}x{\omega }_{0}(x-a)$. Again we have used x_j = ja.

Finally we get the tight-binding model as Eq. (2) in the main text.

Non-Bloch band theory

We have adopted the non-Bloch band theory^3,4,5,6 to investigate the topological properties for case-III with non-Hermitian skin effect. Replacing the vector k by β = e^ik in the Bloch Hamiltonian, the non-Bloch Hamiltonian in the spin space can be written as

$$H(\beta )=\left(\begin{array}{cc}i\gamma -t(\beta +{\beta }^{-1}) & \Omega -{e}^{i{\phi_{0}}}{t}_{so}(\beta -{\beta }^{-1}) \\ \Omega +{e}^{-i{\phi_{0}}}{t}_{so}(\beta -{\beta }^{-1}) & -i\gamma +t(\beta + {\beta }^{-1})\end{array}\right).$$

(12)

Then the eigenvalues E follow

$${E}^{2} = \left[i\gamma -t(\beta +{\beta }^{-1})\right]^{2}+[\Omega -{e}^{i{\phi_{0}}}{t}_{so}(\beta -{\beta }^{-1})]\\ (\Omega +{e}^{-i{\phi_{0}}}{t}_{so}(\beta -{\beta }^{-1})).$$

(13)

For a given value of E, the four solutions of β can be organized as ∣β₁∣ ≤ ∣β₂∣ ≤ ∣β₃∣ ≤ ∣β₄∣. Imposing the condition ∣β₂∣ = ∣β₃∣ would pin down all β-solutions for the generalized Brillouin zone (GBZ).

The non-Bloch winding number accumulated in the GBZ is then

$$W=\frac{i}{2\pi }\int_{\beta }\mathop{\sum}\limits_{\nu =\pm }\left\langle {u}_{\nu L}(\beta )\right|{\partial }_{\beta }\left|{u}_{\nu R}(\beta )\right\rangle ,$$

(14)

where the right and left eigenvectors are defined through $H(\beta )\left|{u}_{\nu R}\right\rangle ={E}_{\beta \nu }\left|{u}_{\nu R}\right\rangle$ and ${H}^{{{{\dagger}}} }(\beta )\left|{u}_{\nu L}\right\rangle ={E}_{\beta \nu }^{* }\left|{u}_{\nu L}\right\rangle$. Note that H(β) satisfies the chiral symmetry σ_yH(β)σ_y = − H(β), for ϕ₀ = π/2, 3π/2.

Topological phase transition point

The gap-closing condition, indicative of the topological transition, requires the solution E = 0 of (13), i.e.,

$$0= {[i{\gamma}_{c}-t({\beta} + {\beta}^{-1})]^{2}+[{\Omega} - e^{i{\phi}_{0}}{t}_{so}({\beta} - {\beta}^{-1})]}\\ {[{\Omega} + e^{{-}i{\phi}_{0}}{t}_{so}({\beta} - {\beta}^{-1})]}.$$

(15)

In combination with the continuum band requirement: ∣β₁∣ ≤ ∣β₂∣ = ∣β₃∣ ≤ ∣β₄∣, we obtain the relation between γ_c and Ω₀, Ω_r. In particular, since the topological zero modes are protected by the chiral symmetry when ϕ₀ = π/2, 3π/2, in the following, we shall discuss the topological transition in these two cases separately.

1. ϕ₀ = π/2

The solution of γ_c as a function of Ω₀, Ω_r can be divided into two regimes

$$(1)\qquad {{{{{{{\rm{for}}}}}}}}\,{\Omega }_{0}\le {\Omega }_{0c}:{\gamma }_{c}={\Omega }_{r};$$

(16)

$$\begin{array}{l}(2)\qquad {{{{{{{\rm{for}}}}}}}}\,{\Omega }_{0}\ge {\Omega }_{0c}:2(t{\gamma }_{c}+{\Omega }_{r}{\Omega }_{0})=(t+{\Omega }_{0})\\ \sqrt{{\left({\Omega }_{r}+{\gamma }_{c}\right)}^{2}+4{t}^{2}-4{\Omega }_{0}^{2}}-(t-{\Omega }_{0})\sqrt{{\left({\Omega }_{r}-{\gamma }_{c}\right)}^{2}+4{t}^{2}-4{\Omega }_{0}^{2}}\end{array}.$$

(17)

Here Ω_0c satisfies

$$(t+{\Omega }_{0c})\left(\sqrt{{\Omega }_{r}^{2}+{t}^{2}-{\Omega }_{0c}^{2}}-{\Omega }_{r}\right)=(t-{\Omega }_{0c})\sqrt{{t}^{2}-{\Omega }_{0c}^{2}}.$$

(18)

2. ϕ₀ = 3π/2

From Eq. (13), we see that the case with ϕ₀ = 3π/2 can be related to that with ϕ₀ = π/2 by the transformation β → β⁻¹. This means that we can directly utilize the four β-solutions in ϕ₀ = π/2 case, i.e., ∣β₁∣ ≤ ∣β₂∣ = ∣β₃∣ ≤ ∣β₄∣, to obtain the solutions in the ϕ₀ = 3π/2 case as ∣β₄∣⁻¹ ≤ ∣β₃∣⁻¹ = ∣β₂∣⁻¹ ≤ ∣β₁∣⁻¹, without any change of the spectrum E. Therefore the topological transition points in the two cases should be identical. We have numerically confirmed that γ_c has the same dependence on the parameters Ω₀, Ω_r as in the case of ϕ₀ = π/2.