FFlow-induced Bulk Fermi Arcs in Simple Elastic Matter
Tsvi Tlusty ∗ Center for Soft and Living Matter, Institute for Basic Science, andPhysics and Chemistry Departments, Ulsan National Institute of Science and Technology, Ulsan 44919, Korea (Dated: February 11, 2021)Open non-Hermitian systems exhibit unique topological hallmarks that are absent from theirisolated Hermitian counterparts. Our model system demonstrates that these exotic topologicalphenomena can be induced in the spectra of generic elastic lattices simply by subjecting them toviscous flow. The interplay of hydrodynamics and elasticity splits Dirac cones into bulk Fermi arcspairing exceptional points with opposite half-integer topological charges. The emergent singularitiesorganize the spectral bands in geometric patterns reflecting the underlying symmetry breaking. Thepresent findings suggest that non-Hermitian topology can be explored in ordinary dissipative matter,opening paths for developing topology-based technology in this regime.
INTRODUCTION
The conservation of energy in isolated Hermitian sys-tems is a basic tenet of physics, but in practice, mostsystems are open, exchanging energy and informationwith the external world. This inherent non-Hermiticityis traditionally seen as an inevitable imperfection ofrealistic systems, yet recent studies revealed that itgives rise to distinctive phenomena unmatched in Her-mitian physics —most notably skewed spectral bandsprone to symmetry breaking when exceptional pointsemerge . This discovery kicked off intensive effortsto engineer non-Hermitian systems in diverse classi-cal and quantum settings, ranging from photonics ,phononics , and optomechanics to electronics and atomic lattices . The potential prowess of non-Hermitian technology has been demonstrated in devel-oping new meta-materials and devices, such as me-dia with loss-controlled transparency or directionalinvisibility , active matter with odd elasticity , andultra-sensitive detectors .Among the exotic phenomena observed in non-Hermitian materials, bulk Fermi arcs hold a specialplace. In contrast to the ingrained intuition that fre-quency levels are closed curves, each Fermi arc is an open isofrequency curve ending at two exceptional points.These endpoints are defects with opposite topologicalcharges of ± . It is important to note that bulk Fermiarcs are topological hallmarks of non-Hermiticity in thebulk spectrum of the lattice, unlike the more familiar sur-face
Fermi arcs induced by Weyl points in 3D Hermitiansystems . Bulk Fermi arcs have been observed so farin one photonic crystal .An open question remains as to whether one can ob-serve and utilize these so-called exotic topological phe-nomena in more common and natural settings. Afterall, we are immersed in a dissipative, non-Hermitianworld, and living systems are immanently open to ex-change with the surrounding environment at all scales. ∗ [email protected] This paper suggests that the answer is positive: simplemodel and simulations show that non-Hermitian topolog-ical hallmarks—in particular, bulk Fermi arcs—can beeasily obtained by tuning the interplay of conservativeand dissipative forces in the low-Reynolds regime, typi-cal to cells, macromolecules and mesoscopic engineeredsystems.
RESULTS
The motion of the hydro-elastic lattice.
To see hownon-Hermitian topology arises in ordinary elastic mat-ter at low-Reynolds, consider the following model system(Fig. 1A). A two-dimensional triangular lattice made ofspherical particles of size l joined by thin elastic struts oflength a and spring constant κ is submerged in a viscousfluid and is moving at a velocity u relative to the fluid.The resulting viscous drag on each particle is γu , where γ is the friction coefficient. Such relative motion can beobtained in the lab by holding the lattice in a flow orby letting it sediment by gravitation in a 2D cell, or bydriving the lattice with various other forces.Perturbing the surrounding fluid, the dragged par-ticles induce long-range interactions throughout thelattice (Fig. 1B). When the flow is limited to a thinsheet of viscous fluid between solid floor and ceiling, thesehydrodynamics forces are dipolar , and the force ex-erted by particle j on particle i is f hyd ij = γ ul R − ij (cid:20) cos 2 θ ij sin 2 θ ij (cid:21) , (1)where R i are the particle positions and the distance vec-tors are R ij = R i − R j = R ij (cos θ ij , sin θ ij ). The mag-nitude of the hydrodynamic dipole in (1) scales as ∼ ul .The Hookean elastic forces are proportional to the changeof the length, ∆ R ij = R ij − ¯ R ij , where ¯ R ij the equilib-rium length of the springs ( a in the lattice), f elas ij = κ ∆ R ij (cid:20) cos θ ij sin θ ij (cid:21) . (2)Owing to the dipolar symmetry of (1), the hydrody-namic forces a pair of particles exert on each other are a r X i v : . [ c ond - m a t . s o f t ] F e b FIG. 1. ( A ) A triangular latticemade of spheres connected by elas-tic struts is immersed in a thin layerof viscous fluid between two walls inthe z direction (perpendicular to thepage). The lattice is driven at a ve-locity u relative to the fluid.( B ) A pair of moving particles inducedipolar flow fields (grey streamlines),thereby exerting on each other equalhydrodynamic forces, f hyd ij = f hyd ji (blue arrows) in the direction of theinduced flow. The elastic forcesalong the elastic strut are opposite, f elas ij = − f elas ji (red arrows), thus con-serving linear momentum. equal, f hyd ij = f hyd ji (because θ ji = π + θ ij ). Thus, thedipolar forces break Newton’s third law of momentumconservation (Fig. 1B). This is because viscous flow isan inherently open system, an effective representationof energy and momentum transfer from hydrodynamicdegrees-of-freedom to microscopic ones. To steadily movethe lattice, the momentum leakage (‘loss’) needs to beconstantly compensated by the driving force (‘gain’). Incontrast, the elastic forces (2) are opposite and conservelinear momentum, f elas ij = − f elas ji . As shown below, theinterplay of conservative and non-conservative forces giverise to skewed non-Hermitian topology.Two timescales govern the dynamics of the lattice: thehydrodynamic timescale τ hyd = a / ( u l ), and the elasticrelaxation time τ elas = γ/κ . The hydroelastic number, (cid:15) ≡ /τ elas /τ hyd = κa γu l , (3)controls the system’s behavior: when (cid:15) (cid:28)
1, it is domi-nated by hydrodynamics, and when (cid:15) (cid:29) τ hyd and frequencies in τ − .The dynamical equations combine Stokes flow withHookean elasticity, a linear regime where the correspon-dence between experiment and theory is well-established.In this overdamped regime, the friction force on each par-ticle is counterbalanced by the driving force, F = F ˆ x ,and by the hydrodynamic and elastic interactions in thelattice, γ ˙ R i = F + (cid:88) j (cid:54) = i ( f hyd ij + f elas ij ) , (4)where ˙ R i is the i th particle’s velocity (Methods). Inequation (4), the long-range hydrodynamic forces (1)are summed over all particles, while elastic interactions(2) are summed only among neighbors connected bystruts. Owing to the lattice parity symmetry, the sumsof interactions vanish at steady-state, when the latticetraverses uniformly at a velocity u = F/γ . Dynamics in momentum space.
The coaction of elas-tic and hydrodynamic forces excites collective motion inthe lattice. Expanding the dynamics (4) in small devi-ations r j of the particles from their steady-state posi-tions in the moving lattice ¯ R j , we find that the collec-tive modes are plane waves r j = | k (cid:105) exp (cid:2) i ( k · ¯ R j − ωt ) (cid:3) (Methods). The 2D polarization of the wave, | k (cid:105) , isan eigenstate of a non-unitary Schr¨odinger-like equationwith an eigenfrequency ω , H | k (cid:105) = ω | k (cid:105) , (5)where the operator H is a momentum-space representa-tion of the forces in the lattice.The “Hamiltonian” H is a 2 × (cid:15) . In the basis of left- and right-circular polariza-tions, we obtain H = H hyd + (cid:15) H elas , with H hyd = Ω x σ x + Ω y σ y , (6)and H elas = − i ( ω x σ x + ω y σ y + ω ) , where σ x , σ y and are Pauli’s and the unity matrices.The frequencies Ω x , Ω y , ω x , ω y and ω , are the Fouriersums of the interactions—all real by the parity symmetryof the lattice (Methods). The hydroelastic operator H (6) is analogous to the Hamiltonian of spin- particles ina complex 2D magnetic field with damping , and thisspinor-like nature shows in the spectrum, as discussedbelow. Symmetry: Parity and Hermiticity.
The hydro-dynamic operator H hyd in (6) is Hermitian, a sum ofproducts of Hermitian Pauli matrices and real numbers.Likewise, the elastic operator H elas is skew-Hermitian( i.e., Hermitian operator multiplied by i ), H hyd = H † hyd , H elas = −H † elas . Note that the hydrodynamic forces do not conserve mo-mentum, but the effective hydrodynamic operator is Her-mitian. This is because, in the low-Reynolds regime, the
FIG. 2. ( A ) A purely hydrodynamic system, (cid:15) = 0. Left: The operator H = H hyd is Hermitian with real frequency bands ω + = − ω − = (cid:112) Ω x + Ω y . The spectrum exhibits six Dirac points (green, one denoted as D) on the boundary of the Brillouinzone (black hexagon). Middle: At each Dirac point, the bands merge, forming a graphene-like double-cone, “diabolo” shape.Right: the 3D shape of the frequency bands in the first Brillouin zone (grey hexagon), showing the double-cones, which arehalved by the zone’s boundary.( B ) At (cid:15) (cid:54) = 0, the symmetry is broken when H includes a skew-Hermitian component, (cid:15) H elas . Left: The real part of the bands,Re( ω + ) = − Re( ω − ), drawn for (cid:15) = π , exhibits six bulk Fermi arcs (white lines) in the first Brillouin zone (the whole spectrumis shown in Fig. S1). Each arc splits from a Dirac point (green) and joins two Exceptional Points (ExPs) with topologicalcharges ± (orange and light blue, E + and E − are two ExPs that split from D ). Middle: the real parts of the bands mergealong the Fermi arc, forming a double-wedge shape. Right: 3D shape of the the frequency bands in the first Brillouin zone(grey hexagon), showing the double-wedges (halved by the zone’s boundary). sum of the forces is proportional to the velocity (the dragforce in (4)). Mathematically speaking, the imaginaryunit factors of the time derivative ( iω ) and the spatialderivative ( i k ) cancel each other. For the same reason,the effective elastic operator is skew-Hermitian, reflectingthe overdamped nature of elastic modes in this regime.As for parity symmetry in k -space, the hydrodynamicpart is odd and the elastic part is even, H hyd ( − k ) = −H hyd ( k ) , H elas ( − k ) = + H elas ( k ) . This follows from the parity of the interactions (1,2): Ω x and Ω y are odd functions of k whereas ω , ω x and ω y are even (Methods, (27,28)). The spectrum, Dirac cones, and bulk Fermi arcs.
The interplay of odd, Hermitian hydrodynamics andeven, skew-Hermitian elasticity brings about distinctivetopological signatures (Fig. 2). The spectrum of equa-tions (5,6) exhibits two eigenfrequency bands, ω ± = − i(cid:15) ω ± √ ν + ν − , (7)where ν ± = (Ω x − i(cid:15)ω x ) ± i (Ω y − i(cid:15)ω y ), and all frequen- FIG. 3. The double branching transition in the real (blue)and imaginary (red) parts of the bands at the exceptionalpoints (10). Zoom on the upper right Fermi arc in Fig. 2B. cies are measured in units of τ − . The correspondingpolarization eigenstates are (Methods) | k ± (cid:105) = 1 (cid:112) | ν + | + | ν − | (cid:20) √ ν ± √ ν ∓ (cid:21) . (8)Without elastic forces (Fig. 2A), a purely hydrody-namic system ( (cid:15) = 0) exhibits real spectrum of propagat-ing phonon-like waves , ω + = − ω − = (Ω x + Ω y ) / .On the edge of the Brillouin zone there are six Diracpoints where the hydrodynamic interaction vanishes,Ω x = Ω y = 0. At a Dirac point, negative and posi-tive bands kiss, ω + = ω − = 0, forming a graphene-likedouble cone .The introduction of skew-Hermitian elasticity at non-zero (cid:15) breaks the symmetry. Fig. 2B shows the realpart of the bands, Re( ω + ) = − Re( ω − ), for a hydroelas-tic number (cid:15) = π (the whole spectrum is shown in Fig.S1). Six bulk Fermi arcs—four S-shaped and two verticallines—emerge from the Dirac points. Along the arcs, thereal parts of the bands merge at Re( ω + ) = Re( ω − ) = 0.Each arc is therefore an open-ended isofrequency contourjoining a pair of isolated exceptional points (ExPs). Asshown below, these points are topological defects of op-posite ± charges.As mentioned, the bulk Fermi arcs are topologicalsignature of non-Hermiticity in the driven elasticlattice, unrelated to he more common surface Fermi arcsinduced by Weyl points in 3D Hermitian systems .As a direct outcome of the broken symmetry whenmixing hydrodynamics and elasticity with oppositesymmetries at (cid:15) (cid:54) = 0, the bulk arcs are a generic feature of the system, observed in other lattice symmetries andfor all directions of the driving flow (Methods, Fig. S2).
Singularities and bifurcations.
At the ExPs, thespectral bands (7) and their corresponding eigenstates(8) simultaneously coalesce (Fig. 2B): the bands arepurely imaginary, ω + = ω − = − i (cid:15) ω , and the coalescingeigenstates, | k + (cid:105) = | k − (cid:105) , are either right- or left-circularpolarizations, reflecting the chirality of the topologicalcharges (Methods). ExPs occur where the determinantin (7) vanishes, ( i.e., when ν + = 0 or ν − = 0) yieldingthe condition, Ω y ω x = − Ω x ω y = ± (cid:15) . (9)At these double branching singularities, of both Re( ω )and Im( ω ), the spectrum becomes gapless. In contrast tothe Dirac points whose eigenspaces are two-dimensional,the eigenstates at the ExPs are parallel, signifying a re-duction of the eigenspace dimension to one.Tuning the hydroelastic number (cid:15) advances the ExPsalong 1D trajectories, from the Dirac points at (cid:15) = 0 tothe corners or the center of the Brillouin zone at (cid:15) = ∞ (Methods). Along these trajectories, the bands exhibitsquare-root singularities (Fig. 3), ω ± = − i(cid:15) ω ± (cid:113)(cid:0) ω x + ω y (cid:1) ( s − (cid:15) ) , (10) FIG. 4. The vorticity of the band-gap ,∆ ω = ω + − ω − (11), for (cid:15) = π . The argument of the band-gap, arg ∆ ω , is color-codedand arrows denote its gradient field, ∇ k (arg ∆ ω ). Hexago-nal black line shows the Brillouin zone boundary, with Diracpoints (green) and exceptional points with ± charges (or-ange and blue). FIG. 5. Left: The density of states g ( ω ) plotted in log-scale, showing the Dirac points (green circles) and the exceptional points(blue/orange circles), all connected by bulk Fermi arcs (white line) at Re ω = 0. Right: 3D representation of g ( ω ). where s ≡ Ω x /ω y = − Ω x /ω y is a 1D coordinate definedby (9). The Fermi arc is the branch-cut of the squareroot (10) stretching between the bifurcation transitionsat the ExPs, s = ± (cid:15) . The square-root singularityreflects strong level repulsion between the bands, com-pared to the linear opening of the gap at the Dirac cone . Topological charges, vorticity and Berry’s phase.
The ExPs are topological defects: a closed eigenfre-quency loop encircling an ExP cannot shrink to a pointwithout passing through the ExP. The correspondingtopological charge can be found from the vorticity of theband-gap , ∆ ω = ω + − ω − (Fig 4), V = − π (cid:73) dk · ∇ k (arg ∆ ω ) . (11)Since the band-gap is ∆ ω = 2 √ ν + ν − (7), the vorticity is V = − π (cid:73) dk · ∇ k (arg ν + + arg ν − ) = ± . (12)with a sign corresponding to the left- or right-handedchiralities of the gradient field (Methods). The vorticityis determined by the branch-cuts, where the jumps ofarg ν ± by ± π yield the topological charges q ± = ± .The charges and their opposite chiralities originate fromthe square-root singularity of the Riemann surface (10),and reflect the spinor-like nature of the polarizationeigenstates, which accumulate a ± π phase when circlingthe ExP and passing through the branch-cut of the arcs. Likewise, the charges can be computed from the inte-gral of the Berry phase (Methods). Berry’s connectionsare the vectors A ± ( k ) = i (cid:104) k ± | ∇ k | k ± (cid:105) , and Berry’s phases γ ± are the loop integrals γ ± = (cid:73) A ± ( k ) · dk = ± π . (13)The corresponding charges, q ± = γ ± / (2 π ) = ± ,are determined by the jump of the phase (13) at thebranch-cut, as in the case of the vorticity (11). The density of states and its singularities.
Project-ing the Riemann surfaces of the spectral bands ω ± ontothe complex frequency-plane reveals the density of states g ( ω ) (Fig. 5), g ( ω ) = (cid:16) a π (cid:17) (cid:90) d k δ ( ω − ω ( k )) . (14)The Dirac points (green) split between the two banks ofthe branch cut. Notable are sharp-edged ridges of thedensity merging at logarithmically diverging summits,akin to van Hove singularities in Hermitian systems .Level repulsion shows in low density of states around theExP singularities in Fig. 5. The analytic density of states g ( ω ) is similar to the one obtained from spectra of sim-ulated lattices, with deviations owing to finite size andboundary conditions (Fig. S3). DISCUSSION
The reported skewed topology with bulk Fermi arcsending at isolated exceptional points was recently discov-ered in a photonic crystal with radiation loss . Here, weproposed a controllable realization of this exotic topol-ogy in ordinary elastic matter, where the analogue of ra-diation loss is viscoelastic dissipation. We demonstratedthat the topology is most notable when the elastic andhydrodynamic forces are comparable, (cid:15) ∼
1, correspond-ing to moderate elastic strengths, κ ∼ ηu ( l/a ) (withStokes law for the friction γ ∼ ηl , where η is the viscos-ity). We conjecture that this regime can be obtained invarious experimental settings and meta-materials, for ex-ample, a crystal of colloidal particles embedded in a gelnetwork, or a lattice of particles connected by springs,sedimenting in a 2D cell. The elastic interaction can bereplaced by other central forces, for example, magneticinteractions. One may speculate that similar topologicalphenomena occur in periodic macromolecular assembliessubject to driving flow.Non-Hermitian topological signatures were recentlyobserved in phononic crystals and in meta-materialswith actively-induced odd elasticity . Composite ma-terials in which such circular active elements are em-bedded in non-circular passive matter can be explored—hypothetically, in systems of cytoskeleton and molecu-lar motors. The ubiquity of ExPs and bulk Fermi arcsin generic elastic matter driven by viscous flow suggeststhat non-Hermitian topology—which is robust againstdefects and noise—can be used to develop devices inthe low-Reynolds regime. Simple viscoelastic systems,such as the one demonstrated here, are proposed as aneasily-accessible playground to investigate basic featuresof topological matter. ACKNOWLEDGMENTS
This work was supported by the Institute for BasicScience, Project Code IBS-R020.
METHODS
Dynamics of hydroelastic lattices.
The following isa brief description of the derivation. An elastic lattice ismoving in the x - y plan of a thin 2D fluid layer at a veloc-ity u relative to the fluid. In this quasi-2D geometry, thenarrow dimension is z (perpendicular to the page in Fig.1). The lattice is made of particles of size l joined bythin elastic rods of average length a and spring constant κ . The viscous drag on each particle is γu , where γ is the friction coefficient (inverse of the mobility). There-fore, this non-equilibrium steady-state of uniformly mov-ing lattice requires driving forces F (force per particle)that inject momentum and energy to compensate for thedissipative friction forces.The particles’ motion with respect to the fluid inducesdipolar perturbations with a velocity field decaying asthe inverse square of the distance, ∼ u ( l/r ) , and thisdipolar flow fields give rise to collective hydrodynamicinteractions . The hydrodynamic force f hyd ij exertedby the j th particle on the i th particle is (in x, y compo-nents), f hyd ij, x = γ Λ (cid:88) j (cid:54) = i (cid:0) X ij − Y ij (cid:1)(cid:0) X ij + Y ij (cid:1) = γ Λ (cid:88) j (cid:54) = i cos 2 θ ij R ij , (15) f hyd ij, y = γ Λ (cid:88) j (cid:54) = i X ij Y ij (cid:0) X ij + Y ij (cid:1) = γ Λ (cid:88) j (cid:54) = i sin 2 θ ij R ij , where the positions of the dipoles are R i = ( X i , Y i ), and R ij = R i − R j = ( X ij , Y ij ) are the distance vectors. Inpolar coordinates, R ij = ( R ij , θ ij ), where R ij = | R ij | and θ ij the angle. Equation (1) is a compact form of(15).The coupling constant Λ scales as the strength of thedipoles, Λ ∼ u l , where l is the size of the particle, and u its velocity relative to the fluid . This defines the typicaltimescale τ hyd of the hydrodynamic interaction, τ hyd ≡ a Λ = a u l , (16)the time it takes a perturbation to traverse a distance a at a sound velocity c s ∼ Λ /a (in the continuum longwavelength limit). The dependence of τ hyd on the physi-cal parameters depends on the forces driving the hydro-dynamic interaction. For example, in a lattice of parti-cles sedimenting in a quasi-2D fluid, the relative velocityscales as u ∼ ∆ ρ gl /η , where ∆ ρ is the density differ-ence and η the fluid’s viscosity. In the quasi-2D flowof squeezed droplets, the coupling is Λ = l Ku , where K = u particle /u fluid ≤ .The elastic Hookean forces are described by harmonicsprings with constant k and equilibrium length ¯ R ij , thelattice constant a . The elastic forces are proportional tothe change of the length, ∆ R ij = R ij − ¯ R ij , f elas i = κ ∆ R ij n ij , (17)where n ij is a unit vector in the direction of the distance¯ R ij . Equation (2) is (17) expressed in polar coordinates.In the overdamped regime, the elastic relaxation time is τ elas = γk . (18)This is the typical decay time of vibrational modes in theabsence of hydrodynamic driving force.The hydroelastic number (cid:15) measures the relative sig-nificance of elastic and hydrodynamic interactions (3), (cid:15) ≡ /τ elas /τ hyd = κa γu l . Unlike odd viscosity or odd elasticity , both hydrody-namic and elastic forces are non-circular, ∇ R ij × f ij = 0,as they conserve angular momentum. Linear Expansion.
In the overdamped low-Reynoldsregime, the equations of motion are (4). At steady-state, the lattice interactions vanish by symmetry, (cid:80) j (cid:54) = i ( f hyd ij + f elas ij ) = 0, and the lattice moves uniformlyat a velocity u = F/γ . Expansion of the equations of mo-tion (4) in small deviations of the lattice positions aroundthe steady-state positions, r j = R j − ¯ R j , yields a lineardynamic equation, ˙ r = H r , (19)where r is 2 N -vector of the N particle deviations, r i .The tensor H = H hyd + (cid:15) H elas , which is analogous to aHamiltonian, combines contributions from hydrodynam-ics and elasticity. H is a 2 N × N -matrix, composed of2 × H ij that account for interactions betweenthe i th and j th particles, H ij = H ij hyd + (cid:15) H ij elas , (20)where the hydrodynamic term is H ij hyd = 2 (cid:18) a ¯ R ij (cid:19) (cid:20) cos 3 θ ij sin 3 θ ij sin 3 θ ij − cos 3 θ ij (cid:21) , (21)and the elastic one H ij elas = 12 (cid:20) θ ij sin 2 θ ij sin 2 θ ij − cos 2 θ ij (cid:21) , (22)which are conveniently expressed in terms of Pauli’s ma-trices σ x , σ y , σ z (and the unity matrix), H ij hyd = 2 (cid:0) a/ ¯ R ij (cid:1) (sin 3 θ ij σ x + cos 3 θ ij σ z ) , (23) H ij elas = ( + sin 2 θ ij σ x + cos 2 θ ij σ z ) . The diagonal terms ensure zero sums, H ii = − (cid:80) j (cid:54) = i H ij .Since the angles obey θ ji = π + θ ij , it follows from (23)that H ji hyd is odd with respect to particle exchange ( i ↔ j ),while H ij elas is even, H ji hyd = − H ij hyd , H ji elas = + H ij elas . The mutual forces between the i th and the j th particlesare f ij = H ij ( r j − r i ) and f ji = H ji ( r i − r j ). Thus,we verify that the hydrodynamic forces violate Newton’sthird law of momentum conservation, whereas the elasticforces obey it (see Fig. 1B), f hyd ji = + f hyd ij , f elas ji = − f elas ij . While the microscopic molecular forces in the fluid obeyNewton’s law, the hydrodynamic interactions are effec-tive macroscopic forces that do not conserve momentum.The momentum is leaking through the walls and is com-pensated by the driving force ( e.g., gravitation or pres-sure gradient).
Momentum space.
To exploit the crystal symmetry,one represents the dynamics in the momentum space ofthe wave-vectors k = ( k x , k y ) by Fourier transform. Thelinearized dynamical equations (19,23) are expanded inplane waves, such that the deviation of each particle fromits mechanical equilibrium position ¯ R j is r j ( t ) = | k ( t ) (cid:105) e i k · ¯ R j = | k (cid:105) e i ( k · ¯ R j − ωt ) , (24)where | k ( t ) (cid:105) = | k (cid:105) exp( − iωt ) is a 2D polarization vectorin k -space.The equation of motion of | k ( t ) (cid:105) is a non-unitarySchr¨odinger-like equation, i ∂∂t | k ( t ) (cid:105) = H | k ( t ) (cid:105) , (25)whose eigenvector is | k (cid:105) with eigenfrequency ω (5), H | k (cid:105) = ω | k (cid:105) . The “Hamiltonian” H is a 2 × H = H hyd + (cid:15) H elas (26)= (Ω x σ z + Ω y σ x ) − (cid:15) i ( ω + ω x σ z + ω y σ x ) . The contributions of the long-range hydrodynamic to H (26) are Fourier sums,Ω x = − i (cid:88) j (cid:54) =0 (cid:18) a ¯ R j (cid:19) cos 3 θ j e i k · ¯ R j , Ω y = − i (cid:88) j (cid:54) =0 (cid:18) a ¯ R j (cid:19) sin 3 θ j e i k · ¯ R j , where ¯ R j = ¯ R j (cos θ j , sin θ j ) are the distances of thesteady-state lattice positions from an arbitrary originparticle 0. Owing to the crystal’s parity symmetry, wecan rearrange the summation to be over pairs at inversepositions, ± ¯ R j , demonstrating that Ω x and Ω y are al-ways real in a crystal, (cid:20) Ω x Ω y (cid:21) = 2 (cid:88) j (cid:54) =0 (cid:18) a ¯ R j (cid:19) (cid:20) cos 3 θ j sin 3 θ j (cid:21) sin (cid:0) k · ¯ R j (cid:1) . (27)The elastic contributions in (26) are sums over the neigh-bors connected by springs to particle 0, ω x = (cid:88) j (cid:54) =0 cos 2 θ j (cid:16) − e i k · ¯ R j (cid:17) ,ω y = (cid:88) j (cid:54) =0 sin 2 θ j (cid:16) − e i k · ¯ R j (cid:17) ,ω = (cid:88) j (cid:54) =0 (cid:16) − e i k · ¯ R j (cid:17) . Again, owing to the crystal parity symmetry, the sumscan be rearranged, demonstrating that ω x , ω y and ω areall real, ω ω x ω y = (cid:88) j (cid:54) =0 θ j sin 2 θ j sin (cid:0) k · ¯ R j (cid:1) . (28)The hydrodynamic and elastic interaction in a triangularlattice are shown in Fig. S4. Spectra.
The eigenfrequencies ω are found by solvingthe secular equation corresponding to (25,26). There aretwo eigenfrequency bands, ω ± = − i(cid:15) ω ± (cid:113) (Ω x − i(cid:15) ω x ) + (Ω y − i(cid:15) ω y ) = − i(cid:15) ω ± √ ν + ν − , (29)where ν ± are defined as ν ± ≡ (Ω x − i(cid:15)ω x ) ± i (Ω y − i(cid:15)ω y ) . In the hydrodynamics-dominated regime, (cid:15) (cid:28)
1, thespectrum (29) is purely real, ω hyd ± ≈ ± (cid:0) Ω y + Ω x (cid:1) / , (30)while in elasticity-dominated regime (cid:15) (cid:29) ω elas ± ≈ − i(cid:15) (cid:104) ω ± ( ω x + ω y ) / (cid:105) . (31)The polarization eignevectors, | k + (cid:105) and | k − (cid:105) , are | k ± (cid:105) = (cid:20) Ω x − i(cid:15)ω x + ω ± + i(cid:15)ω Ω y − i(cid:15) ω y (cid:21) (32)= (cid:20) Ω x − i(cid:15)ω x ± √ ν + ν − Ω y − i(cid:15) ω y (cid:21) , where the eigenvalues ω ± are given in (29). When nor-malized, the eigenvectors become | k ± (cid:105) = 1 √ (cid:112) | ν + | + | ν − | (cid:20) √ ν + ± √ ν − i (cid:0) √ ν + ∓ √ ν − (cid:1)(cid:21) . (33) Circular polarization basis.
One can represent theHamiltonian in the basis of left and right circularly po-larized unit vectors (which are the coalescing eigenstatesat the ExPs), 1 √ (cid:20) ± i (cid:21) . In this basis, the Hamiltonian (26) becomes equation (6), H = H hyd + (cid:15) H elas = (Ω x σ x + Ω y σ y ) − (cid:15) i ( ω + ω x σ x + ω y σ y ) . The eigenvectors in this representation take the simpleform | k + (cid:105) = 1 (cid:112) | ν + | + | ν − | (cid:20) √ ν + √ ν − (cid:21) , (34) | k − (cid:105) = 1 (cid:112) | ν + | + | ν − | (cid:20) √ ν − √ ν + (cid:21) . Dirac points and cones.
Dirac points occur in thepurely hydrodynamic system ( (cid:15) = 0), at wavevectors k D for which the hydrodynamic interaction vanishes,Ω x = Ω y = 0 . From (27), one sees that this happens when sin (cid:0) k · ¯ R j (cid:1) =0, that is for wavevectors that are halves of the reciprocallattice base vectors, b , b , and their combinations, k D = β b + β b (where β , β ∈ {− , , } ). Inthe triangular lattice, the six Dirac points are ( β , β ) =(0 , ± , ( ± , , ( ± , ± dk = k − k D ,Ω x ∼ ∇ k Ω x · dk , Ω y ∼ ∇ k Ω y · dk , where the gradients are ∇ k Ω x = 2 a (cid:88) j (cid:54) =0 ( − β α j + β α j (cid:20) cos θ j sin θ j (cid:21) cos 3 θ j (cid:0) ¯ R j /a (cid:1) , (35) ∇ k Ω y = 2 a (cid:88) j (cid:54) =0 ( − β α j + β α j (cid:20) cos θ j sin θ j (cid:21) sin 3 θ j (cid:0) ¯ R j /a (cid:1) . The α j and α j in (35) are the indices of the lattice points,¯ R j = α j a + α j a , with the basis vectors, a and a . It isstraightforward to verify that the gradients at the Diracpoint are orthogonal, ∇ k Ω x · ∇ k Ω y = 0 , (36)resulting in an elliptic cone (Fig. 2A). It is also useful tonote that, at the Dirac points, the elastic interactions, ω x , ω y and ω , have extrema (minima, maxima, and saddles),and are therefore constant to first order in dk (Fig. S4). Exceptional points and Fermi arcs.
Exceptionalpoints (ExPs) occur when both eigenfrequencies (29) andeigenvectors (33) coincide, ω + = ω − = − i(cid:15)ω , (37) | k + (cid:105) = | k − (cid:105) = 1 √ (cid:20) ± i (cid:21) , where the ± signs correspond to the paired ExPs. In thecircular polarization basis, the eignevectors at the ExPsare the unit vectors, | k + (cid:105) = (cid:20) (cid:21) , | k − (cid:105) = (cid:20) (cid:21) . ExPs are positioned exactly where the determinant, ν + ν − , in (29) vanishes. In other words, when ν + = 0or ν − = 0, which amounts to the double condition (9),Ω y ω x = − Ω x ω y = ± (cid:15) . As one varies (cid:15) , the ExPs move along the 1D curveΩ x ω x + Ω y ω y = 0 , (38)from the Dirac cones at (cid:15) = 0 to the Brillouin zone cor-ners (or its center) at (cid:15) = ∞ . The stretch of the trajec-tory between the two ExPs is the Fermi arc . Along thesetrajectories, the eigenfrequencies are ω ± = − i(cid:15) ω ± (cid:2)(cid:0) Ω x + Ω y (cid:1) − (cid:15) (cid:0) ω x + ω y (cid:1)(cid:3) / (39)= − i(cid:15) ω ± (cid:0) s − (cid:15) (cid:1) / (cid:0) ω x + ω y (cid:1) / , where s ≡ Ω x /ω y = − Ω x /ω y is the coordinate alongthe trajectory, from s = 0, the Dirac point, through s = (cid:15) , the ExP, to the corner of the Brillouin zone s = ∞ (the four S-shaped arcs), or its center (the two straightarcs). The exceptional point is the position of a bifurca-tion transition where the branch-cut of the square rootstretches along the Fermi arc. Bulk Fermi arcs are generic.
The bulk Fermi arcsobserved in the triangular (Figs. 2,S1) or the square (Fig.S2) lattices are a generic phenomenon of the hydroelasticlattice. They emerge as soon as symmetry is broken, at (cid:15) (cid:54) = 0, when the skew-Hermitian elasticity is introduced.To see this formally, one may examine the emergence ofarcs for small (cid:15) . Then, the exceptional point condition(9) can be linearly expended in small deviations from theDirac point, dk = k − k D (35). The double condition (9)becomes a linear equation in dk , (cid:20) ∇ k Ω (cid:124) x ∇ k Ω (cid:124) y (cid:21) dk = (cid:15) (cid:20) ∓ ω y ± ω x (cid:21) , (40)where the ± signs correspond to the two ExPs. Since thegradients are linearly-independent (36), solutions alwaysexist, and the positions of the ExPs are dk = (cid:15) (cid:20) ∇ k Ω x |∇ k Ω x | ∇ k Ω y |∇ k Ω y | (cid:21) (cid:20) ∓ ω y ± ω x (cid:21) . (41)Hence, the arcs and the ExPs are generic. The bulkFermi arc stretches between the two solutions, and itssize grows linearly with (cid:15) . Berry’s phase and the topological charges.
In themomentum representation, Berry’s connections are thevectors , A ± ( k ) = i (cid:104) k ± | ∇ k | k ± (cid:105) , (42) corresponding to the eigenvectors | k ± (cid:105) . For convenience,we write the normalized eigenvectors as Jones polariza-tion vectors, | k + (cid:105) = (cid:20) √ c e iα + √ − c e iα − (cid:21) , | k − (cid:105) = (cid:20) √ − c e iα − √ c e iα + (cid:21) . (43)In this notation, we find that the Berry connections aresimply A ± ( k ) = − c ∇ k α + − (1 − c ) ∇ k α − . (44)For the eignevectors (32), the Jones parameters are c = (cid:12)(cid:12) √ ν + + √ ν − (cid:12)(cid:12) | ν + | + | ν − | ) ,α + = arg (cid:0) √ ν + + √ ν − (cid:1) ,α − = arg (cid:0) √ ν + − √ ν − (cid:1) . The Berry phases γ ± are the path integrals γ ± = (cid:73) A ± ( k ) · dk . (45)To find the Berry phase of the ExPs, we integrate over asmall circular counterclockwise path around each point.For example, consider the point where ν + = 0. In thevicinity of an ExP, c ≈ , and the connections are there-fore, A ± ( k ) ≈ − ∇ k α + − ∇ k α − ≈ − ∇ k arg ν + . (46)Along the circular integration path, the phase arg ν + passes all quadrants. In particular, it passes throughthe branch cut, arg ν + = ± π , where it jumps by ± π ,depending on the gradients of ν + . Thus, we find thatBerry’s phase is γ ± = ± π . Likewise, at the otherExP, Berry’s phase takes the opposite sign, γ ± = ∓ π .Altogether, we find that the corresponding topologicalcharges are q ± = γ ± / (2 π ) = ± . Vorticity and Topological Charges.
Another way tofind the topological charges is through the vorticity of theeigenfrequency band-gap, ∆ ω = ω + − ω − , defined as V = − π (cid:73) Γ dk · ∇ k arg (∆ ω ) , (47)where ω ± are the given in (27). Since, ∆ ω = ω + − ω − =2 √ ν + ν − , we find that the vorticity is V = − π (cid:73) Γ dk · ∇ k (arg ν + + arg ν − ) . (48)As in the case of the Berry phase, the vorticity aroundthe ExPs is determined by their branch-cuts. Namely,the jumps of arg ν ± by ± π when passing through thebranch cuts yield the topological charges q ± = ± .0
1. Bender, C. M. & Boettcher, S. Real Spectra in Non-Hermitian Hamiltonians Having PT Symmetry.
Phys.Rev. Lett. , 5243–5246 (1998).2. Doppler, J. et al. Dynamically encircling an exceptionalpoint for asymmetric mode switching.
Nature , 76–79(2016).3. El-Ganainy, R. et al.
Non-Hermitian physics and PT sym-metry.
Nature Physics , 11–19 (2018).4. Shen, H., Zhen, B. & Fu, L. Topological Band Theoryfor Non-Hermitian Hamiltonians. Phys. Rev. Lett. ,146402 (2018).5. Moiseyev, N.
Non-Hermitian quantum mechanics (Cam-bridge University Press, Cambridge ; New York, 2011).6. Zhou, H. et al.
Observation of bulk Fermi arc and polar-ization half charge from paired exceptional points.
Science , 1009–1012 (2018).7. Feng, L., El-Ganainy, R. & Ge, L. Non-Hermitian pho-tonics based on paritytime symmetry.
Nature Photonics , 752–762 (2017).8. Miri, M. A. & Alu, A. Exceptional points in optics andphotonics. Science (2019).9. Weidemann, S. et al.
Topological funneling of light.
Sci-ence , 311 (2020).10. Rivet, E. et al.
Constant-pressure sound waves in non-Hermitian disordered media.
Nature Physics , 942–947(2018).11. Zhu, W. et al. Simultaneous Observation of a TopologicalEdge State and Exceptional Point in an Open and Non-Hermitian Acoustic System.
Phys. Rev. Lett. , 124501(2018).12. Ma, G., Xiao, M. & Chan, C. T. Topological phases inacoustic and mechanical systems.
Nature Reviews Physics , 281–294 (2019).13. Xu, H., Mason, D., Jiang, L. & Harris, J. G. E. Topo-logical energy transfer in an optomechanical system withexceptional points. Nature , 80–83 (2016).14. Schindler, J., Li, A., Zheng, M. C., Ellis, F. M. & Kottos,T. Experimental study of active LRC circuits with PT symmetries. Phys. Rev. A , 040101 (2011).15. Zhang, Z. et al. Observation of Parity-Time Symmetry inOptically Induced Atomic Lattices.
Phys. Rev. Lett. ,123601 (2016).16. Guo, A. et al.
Observation of PT -Symmetry Breaking inComplex Optical Potentials. Phys. Rev. Lett. , 093902(2009).17. Regensburger, A. et al.
Paritytime synthetic photoniclattices.
Nature , 167–171 (2012).18. Scheibner, C. et al.
Odd elasticity.
Nature Physics , 475–480 (2020).19. Wiersig, J. Enhancing the Sensitivity of Frequency andEnergy Splitting Detection by Using Exceptional Points:Application to Microcavity Sensors for Single-Particle De-tection. Phys. Rev. Lett. , 203901 (2014).20. Hodaei, H. et al.
Enhanced sensitivity at higher-orderexceptional points.
Nature , 187–191 (2017).21. Wan, X., Turner, A. M., Vishwanath, A. & Savrasov,S. Y. Topological semimetal and Fermi-arc surface statesin the electronic structure of pyrochlore iridates.
PhysicalReview B , 205101 (2011).22. Xu, S.-Y. et al. Observation of Fermi arc surface statesin a topological metal.
Science , 294 (2015).23. Yang, B. et al.
Ideal Weyl points and helicoid surfacestates in artificial photonic crystal structures.
Science , 1013–1016 (2018).24. Morali, N. et al.
Fermi-arc diversity on surface termina-tions of the magnetic Weyl semimetal Co Sn S . Science , 1286–1291 (2019).25. Beatus, T., Tlusty, T. & Bar-Ziv, R. Phonons in a one-dimensional microfluidic crystal.
Nature Physics , 743–748 (2006).26. Baron, M., Bawzdziewicz, J. & Wajnryb, E. Hydrody-namic Crystals: Collective Dynamics of Regular Arraysof Spherical Particles in a Parallel-Wall Channel. Phys.Rev. Lett. , 174502 (2008).27. Beatus, T., Shani, I., Bar-Ziv, R. H. & Tlusty, T. Two-dimensional flow of driven particles: a microfluidic path-way to the non-equilibrium frontier.
Chemical SocietyReviews , 5620–5646 (2017).28. Liron, N. & Mochon, S. Stokes flow for a stokeslet betweentwo parallel flat plates. Journal of Engineering Mathemat-ics , 287–303 (1976).29. Cui, B., Diamant, H., Lin, B. & Rice, S. A. AnomalousHydrodynamic Interaction in a Quasi-Two-DimensionalSuspension. Phys. Rev. Lett. , 258301 (2004).30. Landau, L. & Lifshitz, E. On the theory of the dispersionof magnetic permeability in ferromagnetic bodies. Phy.Z. Sowjetunion
8: 153 (1935).31. Yarkony, D. R. Diabolical conical intersections.
Rev. Mod.Phys. , 985–1013 (1996).32. Berry, M. V. Quantal Phase Factors Accompanying Adi-abatic Changes. Proc. Roy. Soc. A , 45–57 (1984).33. Van Hove, L. The Occurrence of Singularities in the Elas-tic Frequency Distribution of a Crystal.
Phys. Rev. ,1189–1193 (1953).34. Avron, J. E. Odd Viscosity. J. Stat. Phys. , 543–557(1998). Supplemental Materials:
Flow-induced Non-Hermitian Topology in Simple Elastic Matter
Additional Figures
FIG. S1.
The spectrum of the triangular lattice.
The real (top) and imaginary (bottom) components of the eigenfre-quencies ω + (left) and ω − (right) of a triangular lattice at (cid:15) = π (Fig. 2B). The black hexagon is the boundary of the Brillouinzone. Shown are the Dirac points (green) at the midpoints of the Brillouin zone edges, and the ExPs with their + (red) and − (blue) charges. Fermi arcs are grey curves. FIG. S2.
The spectrum of a square lattice.
The real (top) and imaginary (bottom) components of the eigenfrequencies, ω + (left) and ω − (right), of a square lattice at (cid:15) = π . The black square is the boundary of the Brillouin zone. Shown are theDirac points (green) and the ExPs with their + (red) and − (blue) charges. Bulk Fermi arcs are grey curves. FIG. S3.
Density of states g ( ω ) : analytic solution vs. simulation. ( A-B ): g ( ω ) for triangular lattice, analytic (A)and simulation (B), for (cid:15) = π . The simulated system is a 121 ×
121 triangular lattice with periodic boundary conditions (atorus). The spectrum is obtained by Fourier analysis of the motion computed from the dynamical equation (4). The simulated g ( ω ) exhibits general similarity to the analytic solution. The incommensurately of the four-fold symmetry of the torus and thesix-fold symmetry of the lattice results in significant deviations. ( C-D ): g ( ω ) for square lattice, analytic (A) and simulation ofa 121 ×
121 lattice (B), for (cid:15) = π . The correspondence between the simulation and the analytic solution is excellent, owing tothe commensurable four-fold symmetry of the lattice and the periodic boundary conditions, with some coarseness due to thefinite size of the simulated system. FIG. S4.
Hydrodynamic and elastic interactions and frequency bands.
The black hexagon is the boundary of theBrillouin zone. The eigenfrequencies of the purely hydrodynamic systems with negligible elasticity, ω hyd ± (30), are real becausethe effective Hamiltonian is Hermitian, H = H hyd . (top-right and center-right). In the absence of flow, the elastic systemis purely damping. H = H elas is skew-Hermitian, and therefore its spectrum, ω elas ± (31) is purely imaginary (top-center andcenter). The elastic interactions, Ω x and Ω y vanish at the Dirac points, the middles of Brillouin zone edges (bottom left andcenter), while the elastic interactions, ω x , ω y and ω1