Topological phase transitions of generalized Brillouin zone

Abstract

It has been known that the bulk-boundary correspondence (BBC) of the non-Hermitian skin effect is characterized by the topology of the complex eigenvalue spectra, while the topology of the wave function gives rise to Hermitian BBC with conventional boundary modes. In this work, we go beyond the known description of the non-Hermitian topological phase and find a different type of BBC that appears in generalized boundary conditions. The generalized Brillouin zone (GBZ) possesses non-trivial topological structures in the intermediate boundary condition between open and periodic boundary conditions. Unlike the conventional BBC, the topological phase transition is characterized by the generalized momentum touching of GBZ, which manifests as exceptional points. As a realization of our proposal, we suggest the non-reciprocal Kuramoto oscillator lattice, where phase slips accompany exceptional points as a signature of such topological phase transition. Our work establishes an understanding of non-Hermitian topological matter by complementing the non-Hermitian BBC as a general foundation of the non-Hermitian topological systems.

Introduction

The rapid progress of non-Hermitian mechanics has paved a new direction to obtain unconventional phases of matter that have no Hermitian analog^1,2,3,4. The representative example is the non-Hermitian skin effect (NHSE), which exhibits the macroscopic localization of bulk states on the boundary. There have been theoretical efforts to understand non-Hermitian bulk-boundary correspondence (BBC) using the bulk topology in the periodic boundary condition (PBC). It has been shown that the winding of complex eigenvalue in one-dimensional systems signifies the presence of the NHSE^5,6,7,8,9. In a more general sense, the macroscopic spectral collapse of the NHSE can be understood within the theory of generalized Brillouin zone (GBZ) defined in the complex plane.

So far, it has been shown that the BBC of non-Hermitian systems is characterized by two distinct types of topology: wave function topology in GBZ and complex eigenvalue spectra topology in conventional BZ. While the wave function topology gives rise to the Hermitian-like boundary modes, the winding number of the eigenvalue spectra in the complex plane signals the presence of the NHSE in the open boundary condition (OBC). Here, to the best of our knowledge, we present the third type of non-Hermitian BBC that emerges in generalized boundary conditions (GBCs). Unlike the case of PBC or OBC, the single wave functions in GBCs possess distinct momenta with multiple GBZs. The existence of multiple GBZ can provoke the intrinsic topological structure of GBZ (Fig. 1), where topological phase transitions are accompanied by exceptional points with touchings of GBZs. The information of the complex eigenvalue spectra itself is ignorant of such concomitant topological phase transition. As a result, our result falls in neither of the two previously known categories of non-Hermitian BBC. In this work, we formulate our theory by presenting the example of the Hatano-Nelson model in GBC. After constructing the single-particle theory, we propose the non-reciprocal Kuramoto lattice as an example of many-body systems. The topological phase transitions between different time-dependent phases are accompanied by the touching of two generalized momenta. Unlike the energy band touching, the touchings of GBZs generally manifest as the coalescence of two eigenstates, known as the exceptional point (EP). The emergence of the EP in the dynamical spectra results in the class of non-equilibrium phase transition, known as exceptional transitions (ET)¹⁰. Our result establishes the important connection between the non-Hermitian BBC and the ET in the non-equilibrium phases of many-body interacting collective behaviors.

Fig. 1: Schematic illustration of non-Hermitian bulk-boundary correspondence in the generalized boundary condition (GBC).

In GBCs, generalized Brillouin zones (GBZs) can possess the non-trivial topological invariant, which manifests as the topological boundary mode. These topological boundary modes are not captured in the conventional band topology. The topological phase transition is characterized by the touchings of GBZs, which manifests as exceptional points.

Results

Non-Bloch band theory in GBC

We start our discussion by considering the Hatano-Nelson model¹¹ with GBCs. The Hamiltonian is given as,

$$\hat{H} = \mathop{\sum }\limits_{i=1}^{N-1}\left[{t}_{R}{\hat{c}}_{i+1}^{{{{\dagger}}} }{\hat{c}}_{i}+{t}_{L}{\hat{c}}_{i}^{{{{\dagger}}} }{\hat{c}}_{i+1}\right]\\ + {t}_{R}^{{\prime} }{\hat{c}}_{1}^{{{{\dagger}}} }{\hat{c}}_{N}+{t}_{L}^{{\prime} }{\hat{c}}_{N}^{{{{\dagger}}} }{\hat{c}}_{1}+{\epsilon }_{1}{\hat{c}}_{1}^{{{{\dagger}}} }{\hat{c}}_{1}+{\epsilon }_{N}{\hat{c}}_{N}^{{{{\dagger}}} }{\hat{c}}_{N},$$

(1)

where t_R(L) represents the hopping amplitude toward the right (left) nearest neighbor site. The second line represents the general deformation of boundary terms, which include the cases of OBC (${t}_{R}^{{\prime} }={t}_{L}^{{\prime} }={\epsilon }_{1}={\epsilon }_{N}=0$), PBC (${t}_{R}^{{\prime} }={t}_{R},{t}_{L}^{{\prime} }={t}_{L},{\epsilon }_{1}={\epsilon }_{N}=0$) and any intermediate boundary conditions between them (See Section IA of supplementary note 1 for full classifications of the boundary conditions).

We represent the eigenstates as, $\left\vert \psi \right\rangle =1/\sqrt{{{{{{{{\mathcal{N}}}}}}}}}\mathop{\sum }\nolimits_{i = 1}^{N}{\psi }_{i}\left\vert i\right\rangle$, where ${\psi }_{i}={\sum }_{\alpha }{c}_{\alpha }{z}_{\alpha }^{i}$ and ${z}_{\alpha }\in {\mathbb{C}}$ is the generalized complex momenta. The eigenstates in the GBC are required to simultaneously satisfy bulk and boundary equations. For Eq. (1), the bulk equation is given by the second-order complex polynomial as,

$$\frac{{t}_{R}}{{z}_{i}}-E+{t}_{L}{z}_{i}=0,$$

(2)

which admits pairs of complex momenta solutions, (z₁, z₂). We call the two momenta (z₁, z₂) paired if they satisfy the bulk equation with the same eigenvalue, E, implying z₁z₂ = r², where ${r}^{2}=\frac{{t}_{R}}{{t}_{L}}$. In addition to the bulk equation, the eigenstates are required to satisfy the boundary equation, ${H}_{{{{{{{{\rm{B}}}}}}}}}{({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}=0$, where

$${H}_{{{{{{{{\rm{B}}}}}}}}}=\left(\begin{array}{cc}A({z}_{1})&A({z}_{2})\\ B({z}_{1})&B({z}_{2})\end{array}\right)$$

(3)

where, $A(z)={t}_{R}-{\epsilon }_{1}z-{t}_{R}^{{\prime} }{z}^{N}$, $B(z)={t}_{L}^{{\prime} }z+{\epsilon }_{N}{z}^{N}-{t}_{L}{z}^{N+1}$. The solutions (z₁, z₂) characterize both the eigenvalues which can now take complex values and the corresponding eigenstates ${\psi }_{i}={\sum }_{\alpha }{c}_{\alpha }{z}_{\alpha }^{i}$. In practice, we determine solutions for (z₁, z₂) by solving Eq. (3), and subsequently solve for the eigenvalue with Eq. (2). Physically, the boundary matrix characterizes the pole structure of the scattering matrix as we consider the boundary as an effective scatterer. (See Sec. IE of supplementary note 1 for the detailed derivation.)

Except for PBC, where the translational symmetry is intact, a wave function is generally described by the two different paired momenta $({z}_{1},{z}_{2})\in {{\mathbb{C}}}^{2}$, which map to the same eigenvalue. The bulk states $\left\vert \psi \right\rangle$ are characterized by the collections of all generalized momenta (z₁, z₂) with z₁z₂ = t_R/t_L which form contours in the complex plane that are called GBZs ${{{{{{{{\mathcal{L}}}}}}}}}_{1}=\{{z}_{1}\},\,{{{{{{{{\mathcal{L}}}}}}}}}_{2}=\{{z}_{2}\}$. Hence, the contour of different momenta constitutes GBZs ${{{{{{{{\mathcal{L}}}}}}}}}_{1}\times {{{{{{{{\mathcal{L}}}}}}}}}_{2}$ due to the constraint z₁z₂ = t_R/t_L¹². For example, in the case of the Hatano-Nelson model, GBZs form two circles with the radii ∣z₁∣, ∣z₂∣ in the complex plane. In this case for OBC, two GBZs overlap each other (∣z₁∣ = ∣z₂∣ = r), which forms a single GBZ^6,7.

In contrast, in the case of GBCs, two GBZs can form two disjoint contours, which is the focus of our study. Generalized momenta z_i outside (inside) the unit circle (∣z_i∣ = 1) or GBZs describes the evanescent wave localized on the left (right) boundary, which corresponds to the boundary states. Touchings of the unpaired momenta of two GBZs correspond to the coalescence of wave functions, which give rise to the EP (See representative examples of GBZs and eigenvalue spectra for different GBCs in Section IC of supplementary note 1).

Topological boundary matrix

The presence of multiple GBZs promotes the spinor representations of the eigenstates. For example, even though the system consists of a single site unit cell, the Bloch wave function can be represented as the expanded spinor ${({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}$, where each component represents the coefficients of the paired momenta z₁ and z₂ respectively. Correspondingly, the boundary equation can be transformed as the eigenvalue equation, ${\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}}{({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}={({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}$, with

$${\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}}=\left(\begin{array}{cc}0&{h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})\\ {h}_{{{{{{{{\rm{B}}}}}}}}}^{-}({z}_{1},{z}_{2})&0\end{array}\right)$$

(4)

where ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})=A({z}_{2})/A({z}_{1})$, ${h}_{{{{{{{{\rm{B}}}}}}}}}^{-}({z}_{1},{z}_{2})=B({z}_{1})/B({z}_{2})$. The transformed boundary matrix ${\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}}$ preserves the chiral symmetry by satisfying the following conditions, $\{{\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}},{\sigma }_{z}\}=0$. The presence of the chiral symmetry indicates that for any right (left) eigenstates $\vert {\chi }_{R(L)}\rangle$ with the complex eigenvalue E(E^*), we can define the right (left) chiral partner states $\vert {\tilde{\chi }}_{R(L)}\rangle \equiv {\sigma }_{z}\vert {\chi }_{R(L)}\rangle$ with the eigenvalue − E( − E^*), which satisfy the bi-orthogonality relations: $\langle {\chi }_{L}| {\chi }_{R}\rangle =\langle {\tilde{\chi }}_{L}| {\tilde{\chi }}_{R}\rangle =1,\langle {\tilde{\chi }}_{L}| {\chi }_{R}\rangle =\langle {\chi }_{L}| {\tilde{\chi }}_{R}\rangle =0$. The chiral symmetry of the effective boundary matrix is responsible for the quantization of the non-Bloch topological invariant in the topologically non-trivial phase which we discuss as follows.

There is a suggestive similarity between the boundary equation [Eq. (4)] and the Bloch Hamiltonian of the non-Hermitian SSH model^6,7. We can assign the topological winding number W_non-Bloch to classify the boundary matrix as,

$${W}_{{{{{{{{\rm{non-Bloch}}}}}}}}}=\frac{{W}_{+}-{W}_{-}}{2},\,\quad \,{W}_{\pm }=\frac{1}{2\pi }{[\arg {h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }]}_{{{{{{{{{\mathcal{L}}}}}}}}}_{1}\times {{{{{{{{\mathcal{L}}}}}}}}}_{2}},$$

(5)

where ${[\arg {h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }]}_{{{{{{{{{\mathcal{L}}}}}}}}}_{1}\times {{{{{{{{\mathcal{L}}}}}}}}}_{2}}$ is the change of phase of ${h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }$ as (z₁, z₂) goes along the GBZ. In general, another non-zero topological invariant, ${W}_{{{{{{{{\rm{non-Bloch}}}}}}}}}^{{\prime} }=({W}_{+}+{W}_{-})/2$, can exist. However, for $({z}_{1},{z}_{2})\in {{{{{{{{\mathcal{L}}}}}}}}}_{1}\times {{{{{{{{\mathcal{L}}}}}}}}}_{2}$, the GBZ ensures that W₊ = −W₋ due to $\det [{H}_{{{{{{{{\rm{B}}}}}}}}}]=0$. As a result, W_non-Bloch can only have non-trivial topological numbers. The physical manifestation of W_non-Bloch manifests as the topological bound state localized on the boundary of the chain (See Methods and also Section ID of supplementary note 1). The meromorphic function $*{h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})\equiv -\frac{{c}_{1}}{{c}_{2}}=-\frac{A({z}_{2})}{A({z}_{1})}=-\frac{B({z}_{2})}{B({z}_{1})}$ are analogous to the reflection amplitudes for the scattering problem in quantum mechanics. Therefore, in order to have a bound state the pole and zero structure of the meromorphic function ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})$ has to change. Hence, the winding number W_non-Bloch defined for the meromorphic function ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})$ will detect the emergence of topological boundary states in the system (See Section IE of the supplementary note 1 for more details).

In the limit of GBC-(I) (${t}_{L(R)}^{{\prime} }=0,\,{\epsilon }_{1},{\epsilon }_{N} \, \ne \, 0$), the boundary matrix H_B becomes Hermitian, where the conventional Berry phase gives the winding number. In other GBCs, ${{{{{{{{\mathcal{L}}}}}}}}}_{1}$ and ${{{{{{{{\mathcal{L}}}}}}}}}_{2}$ can form two disjoint contours in the complex plane. At the topological phase transition of the winding number W_non-Bloch, where ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})$ or ${h}_{{{{{{{{\rm{B}}}}}}}}}^{-}({z}_{1},{z}_{2})$ vanishes, the topological boundary states absorb into the continuum of the bulk bands in the complex momentum plane. Figure 2 shows the topological and exceptional transition of GBZs in the GBC-(I) which we explain in the following manner. In Fig. 2a there are equal number of poles and zeros inside the GBZs and hence the W_non-Bloch vanishes which describes the topologically trivial phase. In Fig. 2b non-zero zero and pole merge with GBZs which describes the exceptional point. In Fig. 2c, there are a pair of poles inside GBZs and hence the W_non-Bloch = 1 which characterizes the topologically non-trivial phase. The merging of the two generalized momenta manifests as the EP, which is signatured by the coalescence of distinct eigenstates and is demonstrated by the vanishing of phase rigidity which is defined as follows

$${R}_{i}=\frac{\langle {\psi }_{i}^{R}| {\psi }_{i}^{L}\rangle }{\langle {\psi }_{i}^{R}| {\psi }_{i}^{R}\rangle },\,i\in \{1,\,2,\,...,\,N\}$$

(6)

where $\vert {\psi }_{i}^{L(R)}\rangle$ are the left (right) eigenmodes corresponding to the eigenvalues E_i. We find that the change in the non-Bloch topological invariant characterizes the number of topological boundary states. We refer to the appearance of the EP during the topological phase transition of the GBZ as the exceptional phase transition. The topological and exceptional phase transitions of GBZs are characterized by the non-Bloch topological invariant W_non-Bloch and phase rigidity R_i which is clearly shown in Fig. 2d–g. We point out that the observed topological boundary mode defies the standard phenomenology of the conventional topological insulators as our model only contains a single band (See Section IF of supplementary note 1 for explicit studies in different GBCs).

Fig. 2: Topological phase transitions of generalized Brillouin zone (GBZ) in generalized boundary condition-(I).

a–c Dictates GBZs ${{{{{{{{\mathcal{L}}}}}}}}}_{1},\,{{{{{{{{\mathcal{L}}}}}}}}}_{2}$ (blue circles and red triangles) and complex vector plot of the off-diagonal element of the boundary matrix ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}$. The winding number associated with the meromorphic function ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}$ determines the non-Bloch topological invariant (W_non-Bloch). d W_non-Bloch vs onsite potential ϵ₁ characterizes topological phase transitions of GBZs at ${\epsilon }_{1}=\pm {t}_{L}r,\,r=\sqrt{{t}_{R}/{t}_{L}}$. e W_non-Bloch in (ϵ₁, ϵ_N)-plane describes the topological phase transition of GBZs. f phase rigidity R₁, R₂ vs onsite potential ϵ₁ characterizes exceptional phase transitions of GBZs at ϵ₁ = ±t_Lr. g total phase rigidity in (ϵ₁, ϵ_N)-plane describes the exceptional phase transition of GBZs. Here t_L (t_R) denotes the hopping amplitude to the left (right) in the bulk of the system, ϵ₁, ϵ_N denotes the onsite potential at two boundaries of the system, and N denotes the system size. The boundary hopping amplitude to the left (right) ${t}_{L}^{{\prime} },\,({t}_{R}^{{\prime} })$ vanishes for generalized boundary condition-(I).

Generalization to many-body systems

We consider the one-dimensional array of the Kuramoto oscillators as a generalization to the many-body systems¹³. In this case, the dynamics of i-th oscillator is described by the phase θ_i(t) which is determined by the following equation,

$$d{\theta }_{i}(t)/dt={K}_{R}\sin (\delta {\theta }_{i+1,i})+{K}_{L}\sin (\delta {\theta }_{i-1,i}).$$

(7)

Here, K_R,L is the coupling with the right and left nearest-neighbor oscillator. δθ_ij = θ_i − θ_j is the phase difference. The model admits the stable limit cycles with the constant phase gradient δθ. Due to the PBC, the phase gradient is quantized as, $\omega =\mathop{\sum }\nolimits_{i = 1}^{N}\delta {\theta }_{i+1,i}/(2\pi )\in {\mathbb{Z}}$. (See Methods and Section II of supplementary note 2 for the exact solutions and stability analysis).

The linearized excitation near the limit-cycle phase is described by the Jacobian of Eq. (7), ${{{{{{{{\mathcal{J}}}}}}}}}_{ij}[\theta ]=-\frac{\partial }{\partial {\theta }_{i}}\left(\frac{\partial {\theta }_{j}}{\partial {\theta }_{t}}\right)$, which is given as,

$${{{{{{{\mathcal{J}}}}}}}}= \mathop{\sum }\limits_{i=1}^{N}{K}_{R}\cos (\delta {\theta }_{i+1,i})\left\vert i+1\right\rangle \left\langle i\right\vert +{K}_{L}\cos (\delta {\theta }_{i-1,i})\left\vert i-1\right\rangle \left\langle i\right\vert \\ - ({K}_{R}\cos (\delta {\theta }_{i+1,i})+{K}_{L}\cos (\delta {\theta }_{i-1,i}))\left\vert i\right\rangle \left\langle i\right\vert .$$

(8)

The Jacobian of the limit cycle phases characterize their local stability and emulates the Hatano-Nelson model with the PBC (Fig. 3a) where the nearest neighbor hopping amplitude is determined by the non-reciprocal couplings (K_R, K_L) and phase differences δθ_i+1, i = 2πω/N. The eigenvalue spectra of the Jacobian form a center-shifted circle in a complex plane comprised of negative real eigenvalues (Fig. 3a). The negative eigenvalue spectrum of the Jacobian characterizes the stability of the limit cycle in non-linear systems^14,15 (See Methods and Section IIC of supplementary note 2).

Fig. 3: Topological phase transitions of generalized Brillouin zones (GBZs) for the Jacobian.

a Dictates GBZs ${{{{{{{{\mathcal{L}}}}}}}}}_{1},\,{{{{{{{{\mathcal{L}}}}}}}}}_{2}$ (blue circles and red triangles) of Jacobian for a chiral phase. b–d Describes the GBZs in generalized boundary conditions for different values of δθ_N between [0, π] which are marked in (i)–(k). e Describes the eigenvalue spectrum of Jacobian for a chiral phase. f–h Describes the eigenvalue spectra in generalized boundary conditions for different values of δθ_N between [0, π] which are marked in (i)–(k). The behavior of GBZs and eigenvalue spectra are similar in the other half. i–k Describes the Real part Re[E], imaginary part Im[E] of eigenvalue spectra, and phase rigidity R_i vs phase difference δθ_N ≡ θ₁ − θ_N during the interval of the phase slip, respectively. At δθ_N = π/2, 3π/2, two GBZs merge defining the exceptional transition which is consistent with the vanishing of the phase rigidity which is shown in panel (k). In the interval of the phase slip for π/2 < δθ_N < 3π/2, the topological boundary state appears with a positive eigenvalue marked with the dashed line in panel (i). Accordingly, we mark the topological boundary state with a circle in panels (d) and (h). Parameters: K_R = 2.5, K_L = 1.9, N = 100. Here K_R (K_L) denotes the nearest neighbor coupling to the right (left), and N denotes the total number of oscillators.

The phase transitions between the limit cycles with different phase gradients (say ω and ${\omega }^{{\prime} }$) are only possible when one of the phase differences undergoes a change of $2\pi (\omega -{\omega }^{{\prime} })$. During the phase transition, a rapid change of the phase difference occurs by $2\pi (\omega -{\omega }^{{\prime} })$, and it is referred to as phase slip where the phase slipping sites is the boundary. In the interval of the phase slip, the phase difference at the boundary δθ_1,N ≡ δθ_N = θ₁ − θ_N changes to δθ_1,N + 2π, and accordingly the boundary hopping amplitudes ${{{{{{{{\mathcal{J}}}}}}}}}_{1,N}={K}_{L}\cos (\delta {\theta }_{N})\,\left.\right({{{{{{{{\mathcal{J}}}}}}}}}_{N,1}={K}_{R}\cos (\delta {\theta }_{N})$ modify. Therefore, in the interval of the phase slip, the Jacobian (Eq. (8)) emulates the Hatano-Nelson model in the generalized boundary conditions. Therefore, in the interval of the phase slip, the Jacobian matrix undergoes the topological phase transition of GBZ which is characterized by the non-trivial winding number. Accordingly, during $\frac{\pi }{2} \, < \, \delta {\theta }_{N} \, < \, \frac{3\pi }{2}$, the topological boundary state is energetically separated from the negative-eigenvalue bulk states and forms the positive real eigenvalue, which physically represents the onset of the limit-cycle instability during the phase transitions (See Methods and Section IIC of supplementary note 2). The adiabatic evolution of GBZs and corresponding eigenvalue spectra are shown in Fig. 3a–d and Fig. 3e–h, respectively, in the half interval of the phase slip (since the evolution is symmetric in the other half). Figure 3i–k shows the variation of the real part of the eigenvalue spectra, the imaginary part of the eigenvalue spectra, and phase rigidity in the interval of the phase slip, respectively.

In addition to the negative eigenvalue bulk states, we identify the Goldstone mode associated with the global U(1) symmetry (θ_i → θ_i + ϕ). Regardless of the phases of oscillators, the Jacobian matrix forms a zero-line sum matrix¹⁶ satisfying $\mathop{\sum }\nolimits_{i=1}^{N}{{{{{{{{\mathcal{J}}}}}}}}}_{ij}=\mathop{\sum }\nolimits_{j=1}^{N}{{{{{{{{\mathcal{J}}}}}}}}}_{ij}=0$. The physical manifestation is the emergence of Goldstone mode, $\vert {\psi }_{{{{{{{{\rm{gs}}}}}}}}}\rangle \sim {(1,1,...,1)}^{T}$ with the zero eigenvalue E = 0. In GBZ, the Goldstone mode touches the singular point, (z₁, z₂) = (1, r²), where ${h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }$ is ill-defined.

At the topological phase transition, $\delta {\theta }_{N}=\frac{\pi }{2},\frac{3\pi }{2}$, the Jacobian exhibits the OBC. The GBZ of the bulk and the topological boundary state meet each other (Fig. 3c). Correspondingly, we observe the coalescence of the Goldstone mode with the normal modes, observed by the vanishing phase rigidity for the first eigenstate. The vanishing phase rigidity can be understood as the consequence of the NHSE where the left eigenstates and the right eigenstates are localized at the opposite boundaries, causing the vanishing wavefunction overlap (See Section IIC of supplementary note 2). This coalescence of the Goldstone mode signifies the non-equilibrium phase transition, which has been referred to as ETs¹⁰. The EP of the Goldstone mode proliferates the temporal fluctuations associated with U(1) symmetry. The physical consequence is the dynamical restoration of the spontaneously broken U(1) rotation symmetry. In other words, in the Hermitian Kuramoto model, the synchronization phase corresponds to the stationary motion (when averaged frequency vanishes) θ_i(t) → θ_i(0) where all the oscillators converge to a specific angle by breaking U(1)-rotation symmetry. However, in the case of the non-Hermitian Kuramoto model, the non-reciprocity produces a finite drift motion that restores U(1)-rotation symmetry.

We propose that the overall chiral velocity serves as a macroscopic order parameter that characterizes such topological non-reciprocal phase transitions,

$${v}_{{{{{{{{\rm{chiral}}}}}}}}}=\mathop{\sum }\limits_{i=1}^{N}\frac{d{\theta }_{i}(t)}{dt}=2\pi \omega ({K}_{R}-{K}_{L}).$$

(9)

The finite chiral motion occurs for each limit cycle due to the non-reciprocity of the couplings. In the previous work¹⁰, a similar chiral motion has been studied in the presence of random noise. Here, we note the crucial difference that the stability of the chiral motion does not require random noise due to the peculiar property coming from the PBC. Instead, the presence of random noise destabilizes the chiral motion by inducing phase slips. During the phase slip, the ET proliferates the Goldstone mode excitations inducing the abrupt change in the chiral velocity (See the time evolution of phases and normalized chiral velocity in the presence of random white noise of strength D in Fig. 4a and b). In the long time limit, the system eventually stabilizes into the limit cycle with the zero winding number (See Methods and Section IIC of supplementary note 2).

Fig. 4: Noise-induced phase transitions.

a Time evolutions of phases in the presence of random white noise of strength D. The dotted lines mark the positions of phase slips. b Time evolution of the total chiral velocity ${v}_{{{{{{{{\rm{chiral}}}}}}}}}^{{\prime} }={v}_{{{{{{{{\rm{chiral}}}}}}}}}/(2\pi ({K}_{R}-{K}_{L}))$. Here K_R (K_L) denotes the nearest neighbor coupling to the right (left). The discrete change of the total chiral velocity serves as the physical observable of noise-induced non-equilibrium phase transition. Parameters: K_R = 2.5, K_L = 1.9, D = 0.15, total number of oscillators N = 100.

Discussion

In general non-Hermitian one-dimensional systems, the one-dimensional non-Hermitian Hamiltonian can be written as,

$$\hat{H}=\mathop{\sum }\limits_{i=1}^{N}\mathop{\sum }\limits_{\alpha ,\beta =1}^{p}\mathop{\sum }\limits_{m=-{n}_{{{{{{{{\rm{L}}}}}}}}}}^{{n}_{{{{{{{{\rm{R}}}}}}}}}}{t}_{m,\alpha \beta }{\hat{c}}_{i+m,\alpha }^{{{{\dagger}}} }{\hat{c}}_{i,\beta },$$

(10)

where α, β ∈ {1, . . . , p} with p being total number of internal degrees of freedom. t_m describes the hopping amplitudes from i-th site to i + m-th site. The Hamiltonian can be represented as (n_L + n_R)-order polynomial, and the wavefunction represented as ${\psi }_{n\alpha }=\mathop{\sum }\nolimits_{i = 1}^{p({n}_{{{{{{{{\rm{L}}}}}}}}}+{n}_{{{{{{{{\rm{R}}}}}}}}})}{c}_{i\alpha }{z}_{i\alpha }^{n}$ is required to simultaneously satisfy the boundary equation, H_B(z_iα)c_iα = 0 with α = 1, . . . , p. Accordingly, each point in the GBZs is described by the effective eigenspinor ${({c}_{11},...,{c}_{1p},{c}_{21},...,{c}_{2p},...,{c}_{({n}_{{{{{{{{\rm{L}}}}}}}}}+{n}_{{{{{{{{\rm{R}}}}}}}}})p})}^{T}$, following the symmetry group of the H_B. The non-trivial homotopy group features the intrinsic topology of the GBZ.

The non-trivial topology of GBZ we reveal here can be readily realized in designable material platforms such as active matter^17,18, metamaterials in the fields of optics and photonics¹⁹, acoustics²⁰, robotics²¹, and electric circuit networks²². In a more general sense, the excitations living on systems with the translational symmetry-breaking defect can be the subject of our study. We have generalized our results to many-body collective behaviors by proposing non-reciprocally coupled oscillators. Various systems described by coupled non-linear oscillators such as active matter, coupled Josephson junctions, and XY spin model belong to this class of dynamical systems^{17,18,23,24,25}. In condensed matter systems, recently proposed studies of spin dynamics with the spin current can be an interesting platform to realize the ET of collective phenomena²⁶.

Methods

Generalized Brillouin zones (GBZs) and their topological structure

The general procedure of finding GBZs and their topological phase transitions in different generalized boundary conditions are discussed in detail in Section I of the supplementary note 1. In this section, we elaborate on our results of the topological phase transition of GBZ for the simplest type of generalized boundary condition (GBC-(I)) defined with ${t}_{R}^{{\prime} }={t}_{L}^{{\prime} }=0,\,{\epsilon }_{1} \, \ne \, 0,\,{\epsilon }_{N}=0$.

Non-Bloch band theory of GBZs in GBC-(I)

In GBC-(I), the eigenvalue equation $\hat{H}\left\vert \psi \right\rangle =E\left\vert \psi \right\rangle$ with $\left\vert \psi \right\rangle =1/\sqrt{{{{{{{{\mathcal{N}}}}}}}}}\mathop{\sum }\nolimits_{n = 1}^{N}{\psi }_{n}\left\vert n\right\rangle$ consists of the bulk as well as boundary equations. The bulk equations are given as,

$${t}_{R}{\psi }_{n-1}-E{\psi }_{n}+{t}_{L}{\psi }_{n+1}=0\,{{{{{{{\rm{for}}}}}}}}\,n\in \{2,...,N-1\}.$$

(11)

We solve the above equations using a single wave vector ansatz of the wave function, ${\psi }_{n}\propto {z}_{j}^{n}$, and the bulk equation transforms to second order complex polynomial equation Eq. (2) which has two solutions z₁, z₂, and regardless of the eigenvalue E, they are related as,

$${z}_{1}{z}_{2}=\frac{{t}_{R}}{{t}_{L}}\equiv {r}^{2},\,\,{{{{{{{\rm{where}}}}}}}}\,\,r=\sqrt{\frac{{t}_{R}}{{t}_{L}}},$$

(12)

In general, the linear superposition of the two ansatz forms general solutions of the bulk equations,

$${\psi }_{n}={c}_{1}{z}_{1}^{n}+{c}_{2}{z}_{2}^{n},$$

(13)

which need to satisfy the following boundary equations in order to be a physical solution,

$${\epsilon }_{1}{\psi }_{1}-{t}_{R}{\psi }_{0}=0,\\ {\psi }_{N+1}=0.$$

(14)

The boundary equations can be written as ${H}_{{{{{{{{\rm{B}}}}}}}}}{({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}=0$, where H_B is given by Eq. (3) with, A(z) = t_R − ϵ₁z, B(z) = z^N+1. We obtain the physical solutions (z₁, z₂) via solving the equation $\det [{H}_{{{{{{{{\rm{B}}}}}}}}}]=0$

$$\frac{{t}_{R}-{\epsilon }_{1}{z}_{1}}{{t}_{R}-{\epsilon }_{1}{z}_{2}}=\frac{{z}_{1}^{N+1}}{{z}_{2}^{N+1}},$$

(15)

together with z₁z₂ = r². The contours of different momenta $({z}_{1},\,{z}_{2})\in {\mathbb{C}}$ represent the GBZs in the complex plane.

For ∣ϵ₁∣ < t_Lr, we find that the two GBZs merge with each other similar to that of OBC (Fig. 2a). For ∣ϵ₁∣ > t_Lr, we also find that the two GBZs merge with each other similar to the previous case, and in addition to that, the boundary equations have one pair of the real solution (z₁, z₂) = (t_R/ϵ₁, r²ϵ₁/t_R) in the large N limit. This pair of disjoint generalized momenta describes the topological boundary state (${\psi }_{n}\propto {z}_{2}^{n}$) localized at the boundary of the chain (Fig. 2c).

Topological phase transition of GBZ

We find that the intrinsic topology of the GBZ can be understood by analyzing the boundary equation which can be transformed as the eigenvalue equation, ${\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}}{({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}={({c}_{1},{c}_{2})}^{{{{{{{{\rm{T}}}}}}}}}$, where ${\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}}$ is given by Eq. (4) with ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}({z}_{1},{z}_{2})=({t}_{R}-{\epsilon }_{1}{z}_{2})/({t}_{R}-{\epsilon }_{1}{z}_{1})$, ${h}_{{{{{{{{\rm{B}}}}}}}}}^{-}({z}_{1},{z}_{2})={z}_{1}^{N+1}/{z}_{2}^{N+1}$. In this case, the transformed boundary matrix ${\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}}$ becomes Hermitian since the meromorphic function ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}$ is unimodular ($| {h}_{{{{{{{{\rm{B}}}}}}}}}^{+}| =1$) together with $\det [{H}_{{{{{{{{\rm{B}}}}}}}}}]=0$, and preserves the chiral symmetry by satisfying the following conditions, $\{{\tilde{H}}_{{{{{{{{\rm{B}}}}}}}}},{\sigma }_{z}\}=0$.

The non-Bloch topological invariant (Eq. (5)) characterizing the topology of the GBZs can be calculated using Cauchy’s argument principle as follows:

$${W}_{\pm }=\frac{1}{2\pi i}{\oint }_{{{{{{{{{\mathcal{L}}}}}}}}}_{1}\times {{{{{{{{\mathcal{L}}}}}}}}}_{2}}\frac{1}{{h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }({z}_{1},{z}_{2})}\frac{d{h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }({z}_{1},{z}_{2})}{d{z}_{1}}d{z}_{1}=Z-P,$$

(16)

where the integration contour ${{{{{{{{\mathcal{L}}}}}}}}}_{1}\times {{{{{{{{\mathcal{L}}}}}}}}}_{2}$ indicates the continuous contour of the GBZs satisfying z₁z₂ = t_R/t_L. Z (P) represents the total number of zeros (poles) inside GBZs counting the multiplicities.

Figures 2a–c and 5a–c show complex vector plots of ${h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }({z}_{1},{z}_{2})$ together with the GBZs and corresponding exemplifies of GBZs together with zeros and poles of ${h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }({z}_{1},{z}_{2})$, respectively. In the topologically trivial phase of GBZ (∣ϵ₁∣ < t_Lr), the interior of the GBZ encloses an equal number of poles and zeros of ${h}_{B}^{+}$ [W_non-Bloch = 0, Figs. 2a and 5a). As the system undergoes the topological phase transition (Figs. 2b and 5b), the zero of ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}$ touches the GBZ. The topological phase transitions of GBZs occur at the exceptional points ϵ₁ = ±t_Lr where the disjoint generalized momenta (z₁, z₂) = (t_R/ϵ₁, r²ϵ₁/t_R) describing the topological boundary state touch the GBZs (Fig. 2a–c). The appearance of the exceptional points is also confirmed by the vanishing of the phase rigidity in the large N − limit (Fig. 2f, g). Finally, for ∣ϵ₁∣ > t_Lr, the GBZ encircles a pair of poles, which manifest as the non-trivial value of the winding number (W_non-Bloch ≠ 0, Figs. 2c and 5c) and describes the topologically non-trivial phase. Moreover, the non-zero zero and pole of ${h}_{{{{{{{{\rm{B}}}}}}}}}^{+}$ coincide with the disjoint generalized momenta in the topologically non-trivial phase and merge with the GBZs at the topological phase transition point.

Fig. 5: Topological phase transitions of generalized Brillouin zone (GBZ) in the generalized boundary condition-I (GBC-(I)).

a–c Examples of the contour of GBZs ${{{{{{{{\mathcal{L}}}}}}}}}_{1},\,{{{{{{{{\mathcal{L}}}}}}}}}_{2}$ together with the location of poles and zeros of the meromorphic functions (${h}_{{{{{{{{\rm{B}}}}}}}}}^{\pm }$) which determine the non-Bloch topological invariant (W_non-Bloch). The arrows beside the zeros and poles indicate the direction of their movement as we tune the parameter ϵ₁. Here, t_L (t_R) denotes the hopping amplitude to the left (right) in the bulk, ϵ₁ denotes the onsite potential at one of the boundaries, and $r=\sqrt{{t}_{R}/{t}_{L}}$.

Exceptional transitions in Kuramoto lattices

We describe the synchronized solutions, their stability analysis, and noise-induced exceptional transitions in detail in Section II of the supplementary note 2. In this section, we briefly describe the methodology with the main results.

Time-dependent synchronized solutions

We consider the locally coupled Kuramoto model with non-reciprocal coupling (K_R ≠ K_L) (Eq. (7)) on a lattice with the periodic boundary condition consisting of N oscillators. This problem is similar to the two oscillators’ problem which hosts time-independent in-phase (for (K_R + K_L) > 0) and out-of-phase (for (K_R + K_L) < 0) synchronized solutions and time-dependent synchronized solutions (for (K_R + K_L) = 0). In this case, the time-dependent synchronized stable phases have been called the chiral phases¹⁰ where two oscillators move with finite common drift velocity $({K}_{R}-{K}_{L})\sin (\delta \theta (0))$ keeping constant phase difference δθ(0) which depends on the initial conditions. For the N-oscillators’ case, we find that the chiral phases exist as a consequence of chase and runaway motion for arbitrary but finite values of (ΔK ≡ K_R − K_L).

We derive the exact form of the synchronized solutions for Eq. (7) with the following ansatz for individual phases

$${\theta }_{i}(t)={v}_{s}t+{\tilde{\theta }}_{i},\,\forall \,i\in \{1,2,...,N\}$$

(17)

here v_s is the common frequency or drift velocity for each oscillator and ${{{\tilde{{{{{\boldsymbol{\theta }}}}}}}}}$ denote constant phases for the oscillators corresponding to the given synchronized solution.

In this case, the common frequency takes the following form (See Section IIC of supplementary note 2),

$${v}_{s}=\frac{{{\Delta }}K}{N}\mathop{\sum }\limits_{i=1}^{N}\sin \delta {\tilde{\theta }}_{i+1,i}\approx \frac{{{\Delta }}K}{N}\mathop{\sum }\limits_{i=1}^{N}\delta {\tilde{\theta }}_{i+1,i}=\frac{{{\Delta }}K}{N}(2\pi w),$$

(18)

for $\delta {\tilde{\theta }}_{i+1,i}\ll 1,\,\forall \,i\in \{1,2,...,N\}$. Here $w\in {\mathbb{Z}}$ is the topological winding number. This result indicates that synchronized solutions of this kind have quantized finite total velocity (in units of 2πΔK) equal to the topological winding numbers $w\in {\mathbb{Z}}$. Suppose ${f}_{i}=\sin \delta {\tilde{\theta }}_{i,i-1}\in [-1,1]$, then Eq. (7) simplifies to

$${f}_{i}=-\frac{{v}_{s}}{{K}_{L}}+\alpha {f}_{i+1},\,\,{{{{{{{\rm{for}}}}}}}}\,\,i\in \{1,2,...,N\}.$$

(19)

with α = K_R/K_L. With the help of the above equation and periodic boundary condition, f_i’s are given by,

$${f}_{i}=-\mathop{\sum }\limits_{j=0}^{N-1}{\alpha }^{j}\frac{{v}_{s}}{{K}_{L}}+{\alpha }^{N}{f}_{i},\,\,{{{{{{{\rm{for}}}}}}}}\,\,i\in \{1,2,...,N\}.$$

(20)

Considering ${f}_{i}={f}_{0}\,({{{{{{{\rm{const.}}}}}}}})\,\forall \,i\in \{1,2,...,N\}$, the above equations implies,

$$\begin{array}{l}{f}_{0}=\frac{{v}_{s}}{{{\Delta }}K}=\frac{1}{N}\mathop{\sum }\limits_{j = 1}^{N}\sin ({\tilde{\theta }}_{j+1}-{\tilde{\theta }}_{j}),\\ \ \Rightarrow \ {\tilde{\theta }}_{i}=\frac{2\pi wi}{N},\,\,{{{{{{{\rm{with}}}}}}}}\,\,\delta {\tilde{\theta }}_{i+1,i}\equiv \delta \theta \approx \frac{2\pi w}{N},\\ {{{{{{{\rm{for}}}}}}}}\,\,\delta {\tilde{\theta }}_{i,i-1}\ll 1,\,\,{{{{{{{\rm{and}}}}}}}}\,\,i\in \{1,2,...,N\}.\end{array}$$

(21)

These synchronized solutions, with quantized total velocity (in units of 2πΔK) proportional to the topological winding number $w\in {\mathbb{Z}}-\{0\}$, and constant phase differences between their neighbors, are known as chiral phases (Fig. 6a, b). The synchronized solution with winding number w = 0 describes the state of complete phase synchronization with vanishing common frequency.

Fig. 6: Schematics of chiral phases.

Vector plots of the time-evolution of phases θ_i(t) in stable chiral phase with winding number (w = 1) for (K_R + K_L) > 0 (a) and for (K_R + K_L) < 0 (b). At t = t_s, the system realizes a given winding number phase, and then at later times, the phases evolve collectively with a common chiral velocity. Here the gray arrows show the previous time orientations of the vectors corresponding to phases θ_i(t), K_R (K_L) denotes the nearest neighbor coupling to the right (left).

Noise-induced exceptional transitions

The Jacobian (Eq. (8)) of the chiral phases emulates the Hatano-Nelson model with the PBC. The eigenvalue spectra of the Jacobian form a center-shifted circle in a complex plane comprised of negative real eigenvalues (Fig. 3a bottom left) which characterize the stability of the chiral phases (See Section IIC of supplementary note 2). We find that the chiral phases are locally stable and the presence of strong and unbounded white noise induces phase transitions among different chiral phases as a consequence of the phase slips. In order to incorporate the white noise term correctly, we implement Euler-Maruyama’s method and also Heun’s method in our numerical simulation to integrate the stochastic differential equation (See Section IIB and IIC of supplementary note 2).

The noise-induced phase transitions can be understood by modeling the adiabatic evolution of the phases as follows:

$${\theta }_{i}(t)=\frac{2\pi wi}{N}(1-t)+\frac{2\pi {w}^{{\prime} }i}{N}t,\,\,(0\le t\le 1),$$

(22)

here w and ${w}^{{\prime} }$ are the winding numbers characterizing different chiral phases. In the interval of the phase slip, one of the phase differences δθ_1,N ≡ δθ_N undergoes a change from δθ_N to $\delta {\theta }_{N}+2\pi (\omega -{\omega }^{{\prime} })$, and the Jacobian realizes the Hatano-Nelson model with generalized boundary conditions. The corresponding boundary matrix for the Jacobian can be described by the Eq. (3) with ${A}{(z)} = {{t}_{R}} - ({t}_{R} - {t}_{R}^{\prime}){z} - {t}_{R}^{\prime} {z}^{N}, {B} (z) = {t}_{L}^{\prime}{z} + ({t}_{L} - {t}_{L}^{\prime}) {z}^{N} - {t}_{L} {z}^{N + 1} , \, {t}_{R}^{\prime} = {t}_{R} \cos (\delta \theta_{N}) , \, {t}_{L}^{\prime} = {t}_{L} \cos (\delta \theta_{N})$. Accordingly, at δθ_N = π/2, 3π/2, the Jacobian realizes OBC where two GBZs merge (Fig. 3c). Correspondingly, we observe the coalescence of the Goldstone mode with the normal modes (exceptional point), observed by the vanishing phase rigidity for the first eigenstate as a consequence of the NHSE for the Jacobian which is referred to as exceptional transition. Also, during $\frac{\pi }{2} < \delta {\theta }_{N} < \frac{3\pi }{2}$, the non-trivial topology of GBZ which is characterized by the non-trivial winding number leads to the positive real eigenvalue topological boundary state (Fig. 3d) (See Section IIC of supplementary note 2).