Realization and topological properties of third-order exceptional lines embedded in exceptional surfaces

As the counterpart of Hermitian nodal structures, the geometry formed by exceptional points (EPs), such as exceptional lines (ELs), entails intriguing spectral topology. We report the experimental realization of order-3 exceptional lines (EL3) that are entirely embedded in order-2 exceptional surfaces (ES2) in a three-dimensional periodic synthetic momentum space. The EL3 and the concomitant ES2, together with the topology of the underlying space, prohibit the evaluation of their topology in the eigenvalue manifold by prevailing topological characterization methods. We use a winding number associated with the resultants of the Hamiltonian. This resultant winding number can be chosen to detect only the EL3 but ignores the ES2, allowing the diagnosis of the topological currents carried by the EL3, which enables the prediction of their evolution under perturbations. We further reveal the connection between the intersection multiplicity of the resultants and the winding of the resultant field around the EPs and generalize the approach for detecting and topologically characterizing higher-order EPs. Our work exemplifies the unprecedented topology of higher-order exceptional geometries and may inspire new non-Hermitian topological applications.

may even give rise to non-Abelian topological charges [4,5].Nodal surfaces have also been shown to carry topological charges [6,7].Recently, physicists found that introducing non-Hermiticity further enriches the diversity of band topology [8][9][10][11].This is partly due to the fact that the non-Hermitian spectrum occupies the complex plane, such that the energies themselves can exhibit topological winding behaviors, leading to an additional layer of "spectral topology" underneath the wavefunction topology that is studied for Hermitian systems, giving rise to skin effects [12][13][14][15][16] and spectral knots [17,18].Non-Hermitian degeneracies known as EPs possess topological properties characterizable by spectral winding numbers [8,10,11,[19][20][21].In most studies, EPs are formed by two coalescing states.These are called order-2 EPs, and akin to Hermitian degeneracies, they can also form nodal structures, such as rings [22][23][24], lines [25][26][27], links and chains [25,27,28], and surfaces [29,30].Higher-order EP is formed when three or more states coalesce.They are intrinsically more stringent to realize, and their stable existence demands more degrees of freedom in the parameter space or a higher level of symmetries [31,32].For these reasons, higher-order EP structures are so far beyond reach experimentally and are less wellunderstood theoretically.
Here, we report the experimental realization of EL3 entirely embedded on ES2.Both the EL3 and ES2 run continuously through the entire 3D parameter space, which is homeomorphic to a 3torus by design.Such geometry presents an unexpected difficulty for topological characterization.
The prevailing method extracts topological properties of nodal degeneracies by evaluating topological invariants either on a 2-sphere enclosing the entire nodal structure, with the topological charges of a Weyl point being an important case; or on a 1-sphere encircling the nodal structure, such as the characterization of topological nodal lines [3,[7][8][9]33] and order-2 EP lines [21,24].
(Topological categorization using 0-sphere, i.e., two separate points, is also possible but not widely adopted.Hence it is not our focus here.)However, in our case, any enclosing sphere of a single EL3 would encounter ill-behaved spectral singularity on the ES2, thus defying the continuous requirement for spectral winding.This conundrum is resolved by exploring the resultants of the Hamiltonian matrix [34], which are auxiliary manifolds associated with but different from the eigenvalue manifolds.A "resultant vector field" can be chosen to vanish only at the EL3 and remain continuous at the ES2, through which a resultant winding number can be computed for diagnosing the topology of the EL3 while ignoring the influence of the ES2.The validity of our approach is verified by successfully predicting the local evolution of a touching point (TP) of two EL3 under perturbation.Our study expands the understanding of non-Hermitian topology by unveiling novel topological scenarios exclusive to higher-order EP structures.
Realization of symmetry-protected EL3.-First, we present an experiment-feasible lattice system that realizes the said EL3.We begin with a codimension analysis of an order- EP (EP), i.e.,  -fold non-Hermitian degeneracy point.An isolated EP  requires 2( − 1) degrees of freedom (DOFs) to have a solution.In the absence of any symmetry, a -dimensional structure constituted by EP lives in the parameter space with minimal 2( − 1) +  dimensions.Hence both isolated EP3 and ES2 are stable in a four-dimensional (4D) parameter space [30,31].Higherdimensional systems have been studied in diverse scenarios using synthetic dimensions, such as acoustics [35][36][37], photonics [38][39][40], and electric circuits [41,42], they nevertheless tend to be more difficult to handle and realize.Fortunately, the dimensionality requirement can be reduced by enforcing additional symmetries [31].In particular, when parity-time symmetry is respected, the characteristic polynomial of a Hamiltonian , denoted () = det( − ) = 0 where  denotes the eigenvalues and  is an identity matrix, has entirely real discriminant  ∈ ℝ, i.e., Im  = 0 is always satisfied.Hence the DOF requirement is reduced from 2( − 1) to  − 1.In other words, the codimension of EP2 is reduced by one and the codimension of EP3 by two [31,43].
Consequently, both ES2 and EL3 are accessible in a 3D PT-symmetric three-state system, serving as our starting point in designing an experiment-feasible lattice model.
We base our experimental system on coupled acoustic cavities [37,44].Here, we further engineer the system such that its parameter space is mapped to a 3D lattice model.We begin with three air-filled cylindrical cavities stacked together [Fig.1(a)].Within each cavity, a thin plate is fixed in the radial direction, which reflects the circulating propagative waves and thus leads to the formation of azimuthal standing-wave modes.We use the second azimuthal mode, whose pressure  and velocity  profiles are shown in Figs.1(b) and 1(c), respectively.The mode is harmonic in the azimuthal angle , with a pressure node (velocity anti-node) along the diameter perpendicular to the plate.Such mode profiles inspire us to use the azimuthal angle to realize three 2-periodic synthetic coordinates, denoted ( 1 ,  2 ,  3 ) .Because the parameter space is clearly a homeomorphism of a 3-torus, we call it a 3D synthetic Brillouin zone (SBZ) henceforth.The azimuthal position of the nodal line is set as the "origin" of the three coordinates.Let  1 tune the imaginary part of onsite resonant frequency in cavities A and C, which is the source of non-Hermiticity in our system.This is achieved by placing a piece of acoustic sponge to generate losses, which are linearly proportional to the local kinetic energy, given as  ∝  2 ( 1 ) ∝ sin 2  1 .Let  2 modulate the real part of the onsite resonant frequency of cavity B. A small metallic block is placed on the circumference for this purpose, and its azimuthal position is assigned as  2 .Its perturbation to the resonant frequency is linear to the local acoustic potential energy  ∝  2 ( 2 ) ∝ cos 2  2 [45].Assign  3 to control the coupling strength between cavities A and B, also B and C. Acoustic coupling strength is proportional to pressure intensity, i.e.,  2 ( 3 ) ∝ cos 2  3 .
To satisfy PT symmetry, an equal amount of acoustic sponge to all three cavities as the biased loss, and then a specific amount of sponge in cavity A is relocated to C, such that an effective gain is created in A and the same amount of loss is added to C. We have characterized the detuning, loss, and coupling in our systems, and the results are shown in Fig. 1(d-f) as functions of the corresponding synthetic momenta.By delicately tuning the acoustic parameters of the whole ternary cavity system, we obtain that the tunable loss in the system follows ( 1 ) = −60.86× (0.50 sin 2  1 − 1), the detuning of cavity B is described by ( 2 ) = −26.60cos 2  2 , and the coupling between neighboring cavities obeys ( 3 ) = −42.91 × (1 − 0.90 cos 2  3 ).
Our acoustic design is captured by a three-state Hamiltonian  = ( 0 −  0 ) +  3 , where  0 is the resonance frequency of the second azimuthal cavity mode and  0 is the dissipation rate.
The second term is ).
( The emergence of an EP can be identified by the conditions ℛ[ () ,  (+1) ] = 0 with 0 ≤  <  − 1 , and ℛ denotes the resultant,  () is the  th-order derivative of the characteristic polynomial with respect to .As such, we identify both ES2 and EL3 in the SBZ, as shown in Fig. and the TPs will be discussed later.
The ES2 and EL3 are observed in our acoustic experiments.To this end, the acoustic pressure responses at each cavity are measured near  0 at different synthetic momenta.The real and imaginary parts of the eigenfrequencies are then extracted from the acoustic responses using the Green's function [37,44,46] (see Supplementary Materials, Section II for details).We fix  2 = 0.5, then choose five different  3 indicated by the horizontal dashed lines in Fig. 3(a), and for each  3 , the acoustic system is tuned to five different  1 .The real parts of the eigenfrequencies from the measured data are depicted in Fig. 3(b), which shows good agreement with the theoretical results (solid curves).Therein, the EP3 can be easily identified at positions where all three eigenfrequencies converge.These positions are then marked in Fig. 3(a) with the stars and fall on the computed EL3.We then observe the ES2 by performing similar experiments in different  1  2planes at  3 = 0.5 and  3 = 0.33, which intersect with the ES2 and the EL3.The coalescence of two of the three states or all three states is clearly seen in Fig. 3(d, f), and the measured locations of the EP2s and EP3s also conform well with the theoretical results.] ≠ with ,  indexing the states, which is a topological invariant for the spectral topology of the eigenvalue manifold.The result is  = 0, which does not reveal the topological details carried by the ES2 or EL3 individually.
We thus must find an approach to characterize the topology of the EL3, which seemingly refuses any prevailing characterization methods.To its resolve, recall the discriminant of (), −ℛ[,  (1) ] rules out ℛ[,  (1) ], and the other two resultants, ℛ[,  (2) ] and ℛ[ (1) ,  (2) ], shall be used because they fulfill the above requirement.Because generically ℛ[ () ,  () ] ∈ ℂ, the resultants themselves form a complex manifold that complements the eigenvalue manifolds, and there is a one-to-one correspondence between the zeros on the resultant manifold with EPs [34].
In Fig. 4(b), we plot the Λ as a vector field on the  1  2 -plane at  3 = 0.7 [the green plane in Fig. 4(a)], which intersects with two EL3.Λ is indeed vanishing at the EP3, but it does not generate any vortex.Consequently, the winding numbers of Λ, defined as are zero for both EP3, suggesting that they both are unstable.To further reveal the local evolution of the EP3, we introduce two types of symmetry-preserving perturbations (  and   ): ( 1 ) = −60.86(1− 0.50 sin 2  1 +   ) and ( 2 ) = −26.60cos 2  2 +   .When perturbation is off, i.e.,   =   = 0, the two surfaces defined by  = 0 and  = 0 are respectively shown by orange and blue surfaces in Fig. 4(a).According to Bezout's theorem, the number of intersection points is four in  1  2 -plane when considering both the complex domain and intersection multiplicity.
However, in Fig. 4(b), only two intersections are found, which are the EP3s.This indicates a twofold multiplicity for both intersections.Further changing  3 to /2, two EL3 merge and form the TP, which clearly has a multiplicity of four.
The two-fold multiplicity of the EL3 combined with their vanishing  Λ together suggests that the EL3 in Fig. 4(a) can be made locally stable without breaking the symmetry.The speculation is easily verifiable by letting either   or   be non-zero.Figure 4(c) shows that when   = −0.1, the EL3 split into two pairs symmetric about the  2 = /2plane, and they do so without dropping the order.Figure 4(d) plots the Λ-field and the solutions for  = 0,  = 0 in the  1  2 -plane at  3 = 0.7.Clearly, four EP3 are seen, indicating the removal of multiplicity.The EP3s can be separated into two pairs by the opposite vortices they carry, indicating  Λ = ±1.In other words, the EL3s are now topologically stable.Note that the TPs from the crossings of the two oriented order-3 ELs possess zero  Λ , and their multiplicity is reduced to two.We can use  Λ to assign each EL3 with a "topological current," as indicated by the arrows in Fig. 4(c).Indeed, the currents cancel when the two pairs of EL3 merge.Herein, an alternative description is that EL3 carrying opposite topological currents can merge without annihilation, giving rise to higher-order EL that are "topologically neutral."Such a scenario, which resembles the accidental degeneracy appearing in Hermitian band structures, has not been reported before.
The topological currents are also informative in revealing the local evolution of the TPs.Such well-behaved at the ES2 and only detects the EL3.Our findings show that higher-order EPs possess far richer topological properties that are absent for both EP2 and Hermitian degeneracies.The exploration of these properties may lead to new phenomena and applications relating to non-Hermitian energy transfer [47,48] or wave manipulations [49].
Equation ( 6) can also be derived from the formulas of complex eigenvalues of the Hamiltonian.
Herein,  contains the information of both EP2s and EP3s.To further distinguish EP3s from EP2s, he need to find the exclusive condition for EP3s.

II. Retrieval of eigenfrequencies
The eigenfrequencies are obtained from the measured response spectra by utilizing the Green's function [3,4], given by where   denotes the eigenfrequency with  labeling the eigenstates.

2
(b).The real-eigenfrequency Riemann surfaces on three distinct 2D slices are displayed in Fig.2(c-e).Panels (d) and (e) show  1  2 -planes sliced at  3 = /2 and  3 = /3, respectively.EL2 are observed as the consequence of the slicing planes cutting across the ES2, as highlighted by the dashed blue curves.The remaining state (shown in orange) touches the EL2 at particular isolated points and forms EP3 (purple hexagons and red stars).These EP3 only appear when  2 = ±/2, or equivalently ( 2 ) = 0 .The conditions for EP3 to appear are ( 2 ) = 0 , ( 1 ) ± √2( 3 ) = 0, where the ± sign suggests two possible solutions (see Supplementary Materials, Section I for details).The EL3 are plotted in Fig.2(b).They are also seen in Fig.2(c), which plots the  1  3 -plane at  2 = /2.Herein, the EL3 forms a linear crossing at  1 = 0, ±, which we denote as touching points (TPs) and mark by the purple hexagons.The characteristics of the EL3 as  = ∏ (  −   ) < .The equation  = 0 indicates two or more identical eigenvalues and is commonly used as the condition for identifying an EP  .However, importantly, this condition does not distinguish .The ensuing winding of , called the discriminant number widely used to characterize EP2, is equivalent to , which reflects the intersection multiplicity of Re() = 0 and Im() = 0[20].This enlightens that the quantity defined to model the EL3 shall vanish only at the EP3 to avoid the influence of the inevitably surrounded EP2s.The fact that  =

a
configuration discloses two possible local evolutions in the natural projective plane ( 1  3 plane).When the TP is open, the two linear-crossed EL3s can only separate without violating the orientation defined by the currents.The two possible cases are shown in Figs.4(e) and 4(f).Discussion and conclusion.-Wehave experimentally realized EL3 embedded in ES2 across the entire 3D SBZ.The discovery of such unconventional exceptional geometry reveals a multitude of intriguing aspects of higher-order EP previously unknown.The topological characterization of the EL3 demands escaping to an auxiliary resultant manifold, which remains

FIG. 1
FIG.1 Experimental realization of the synthetic space.(a) A ternary acoustic cavity system realized the synthetic space.Each cavity has a height of 4 cm and a radius of 5 cm.(b-c) Acoustic pressure (b) and velocity field (c) of the second-order mode for a single cavity.(d-f) The onsite detuning, hopping strengths, and onsite loss as functions of  2 ,  3 , and  1 , respectively.The blue curves are fitted from experimental data (red circles).
FIG. S1.Selected results of the measured and retrieval acoustic pressure response spectra.Red open circles are experimentally attained data.Blue curves are fitted by applying the Green's function method.Good agreement betheen the tho indicates the validity of our fitting approach.
is also the condition of EP2 and has singularities at the positions of EP3s.Hence one cannot use   to characterize the multiplicity of the EP3.To further demonstrate, he define tho different resultant fields Λ ′ =  + , and Λ ′′ =  + .They are plotted in Fig. S2(b) and (c).

Fig. S2 .
Fig. S2.The resultant fields (arrohs) in the  1  2 -plane at  3 = 0.7, as denoted by the green plane labeled in Fig. 4(a).Here, the fields are Λ =  +  for (a), Λ ′ =  +  for (b), and Λ ′′ =  +  for (c).The evolutions of the argument hhen encircling the EP3s along the purple closed loops in (a-c) are plotted in (d-f).In (a-c),   ,   and   are represented by blue, red and green Characterization of the EL3.-The presence of both ES2 and EL3 gives rise to intriguing topological characteristics.The ES2 form close, continuous 2D surfaces that kiss at the TPs, which separate the eigenvalue manifold into disjoint regions.The EL3 are entirely embedded in the ES2 and also osculate at the TPs.Also, both the ES2 and EL3 run through the entire SBZ in the  1 -that no 2-sphere can enclose them.Yet it remains possible to encircle both the ES2 and EL3 with a 1-sphere.We have computed the eigenvalue winding number, defined as  = direction.Such peculiar geometries entail difficulties in their topological characterization.As mentioned before, the topology of a nodal structure is diagnosed by invariants computed on the -spheres with 0 ≤  ≤  − 1, which enclose the nodal structures.Examining the ES2 and EL3, it is clear 1* 1 Department of Physics, Hong Kong Baptist University, Kowloon Tong, Hong Kong, China 2 Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 3 Department of Physics, State Key Laboratory of Surface Physics, and Key Laboratory of Micro and Nano a). Hence the multiplicity of the intersection is precisely the multiplicity of the EP3.For our Hamiltonian  3 ,  and  are (22)tituting Eq. (21) into Eq.(19)gives the location of EL3s 0.50 sin 2  1 − 0.90 cos 2  3 =   ,(22)0.50sin 2  1 + 0.90 cos 2  3 = 2 +   .(23)  , and  , .From Eq. (2) and Eq.(9), he can deduce  = Remarkably, the tangent of   , denoted as (  ), is  2 , hhich is of multiplicity 2.   also has one tangent of  2 hith the multiplicity being 1, but   =  1 does not have any linear term in  2 , leading to (  ) = 0. Then he have   , (  ,   ) = (  ) + (  ) = 3, (31a)   , (  ,   ) = (  ) + (  ) = 2, (31b)   , (  ,   ) = (  ) + (  ) = 1.(31c) Apparently, only   , (  ,   ) exactly characterize the multiplicity of EP3.This is because