Non-Hermitian band topology from momentum-dependent relaxation in two-dimensional metals with spiral magnetism

We study the emergence of non-Hermitian band topology in a two-dimensional metal with planar spiral magnetism due to a momentum-dependent relaxation rate. A sufficiently strong momentum dependence of the relaxation rate leads to exceptional points in the Brillouin zone, where the Hamiltonian is nondiagonalizable. The exceptional points appear in pairs with opposite topological charges and are connected by arc-shaped branch cuts. We show that exceptional points inside hole and electron pockets, which are generally present in a spiral magnetic state with a small magnetic gap, can cause a drastic change of the Fermi surface topology by merging those pockets at isolated points in the Brillouin zone. We derive simple rules for the evolution of the eigenstates under semiclassical motion through these crossing points, which yield geometric phases depending only on the Fermi surface topology. The spectral function observed in photoemission exhibits Fermi arcs. Its momentum dependence is smooth -- despite of the nonanalyticities in the complex quasiparticle band structure.

Introduction. The discovery of topological insulators [1,2] has triggered a systematic analysis and classification of topological features of band structures in solids. So far, the main focus has been on noninteracting electrons and superconductors in a mean-field picture, corresponding to Hermitian quadratic Hamiltonians [3]. Recently, there has been growing interest in topological features of non-Hermitian Hamiltonians [4]. In quantum manybody systems, these naturally arise in certain open systems, but also as effective Hamiltonians capturing relaxation processes in interacting and/or disordered closed systems [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. A striking effect, which is unique to non-Hermitian systems, is the existence of exceptional points in momentum space where the Hamiltonian is not diagonalizable.
In this letter we show that the combination of two seemingly innocuous ingredients -spiral magnetic order in a two-dimensional metal and a momentum-dependent relaxation rate -can lead to a non-Hermitian Hamiltonian with non-trivial topological features, such as exceptional points and branch cuts in the Brillouin zone. Spiral order is a candidate for incommensurate magnetic order observed in cuprate superconductors [20][21][22][23][24][25][26][27][28][29][30], while relaxation rates with a pronounced momentum dependence arise naturally in two-dimensional systems with strong antiferromagnetic fluctuations [31][32][33][34]. We find that the momentum dependence of the relaxation rate can lead to a closing of the direct band gap between the quasiparticle bands E + p and E − p on one-dimensional branch cuts in the Brillouin zone, which terminate at exceptional points. These lines of degenerate quasiparticle bands, which are sometimes referred to as non-Hermitian (bulk) Fermi arcs [4,5], are in general dispersive, in contrast to flat degenerate bands in some other systems [5,16,35]. In the dispersive case, hole and electron pockets merge at isolated momenta in the Brillouin zone where these degenerate bands cross the Fermi level, leading thus to a pe-culiar Fermi surface topology. Electrons traversing such crossing points along the Fermi surface acquire π-phase shifts, which can lead to a non-trivial geometric Berry phase. Surprisingly, we find that the non-analyticity of the complex band at the exceptional points does not entail any singularity in the spectral function for single electron excitations. Nevertheless, the Fermi surface obtained from the spectral function seems truncated to Fermi arcs. Spiral spin density waves. In a planar spiral spin density wave, the local magnetic moment has the form S i = mn i , where m is a constant amplitude and n i a site-dependent unit vector, which rotates in a fixed but arbitrary plane. For definiteness, we choose n i to lie in the x-y plane, such that n i = cos(Q·R i ), sin(Q·R i ), 0 , where Q is the wave vector of the spin density wave. On a mean-field level, the planar spiral spin density wave is described by the two-band tight-binding Hamiltonian H = p Ψ † p H p Ψ p , where the spinor Ψ p = (c p+Q,↑ , c p,↓ ) collects the two spin components with a relative momentum shift Q, and where p is the (bare) band dispersion and ∆ the magnetic gap [36]. The wave vector Q can be chosen arbitrarily. The simple two-band structure is due to a symmetry under combined lattice translations and spin rotations [37]. For Q = (0, 0) and Q = (π, π) one recovers ferromagnetic and Néel antiferromagnetic order aligned in the xy plane, respectively. In the following we consider incommensurate spiral order with wave vectors of the form Q = (π − 2πη, π) with η > 0, as found in the hole-doped Hubbard and t-J model [20][21][22][23][24][25][26][27][28][29][30]. Diagonalizing H p one finds two quasi-particle bands E ± p with a minimal direct gap ∆, which generally results in a reconstructed Fermi surface with electron and/or hole pockets [28][29][30].
Non-Hermitian effective Hamiltonian. In interacting electron systems all the information on the Fermi surface, quasi-particle bands and decay rates, as well as the spectral function measured in photoemission, is encoded in the single-particle Green's function. The bare Green's function of the noninteracting reference system is dressed by the self-energy, which receives contributions from the electron-electron interaction, and possibly from phonon and impurity scattering. In the low-frequency limit, the real part of the self-energy yields a renormalization of the band structure, and a reduction of the quasiparticle weight, while the imaginary part describes the quasiparticle relaxation rate Γ p . Here we discard the real part and focus on the more interesting effects of Γ p in combination with spiral magnetic order. In the two-component spinor basis defined above, the retarded Green's function can then be written as where the non-Hermitian Hamiltonian H p is defined as with H p from Eq. (1). Imaginary off-diagonal components have only minor consequences [38]. Exceptional points. H p has the complex eigenvalues with the discriminant where s/a p = 1 2 ( p+Q ± p ) and Γ s/a p = 1 2 (Γ p+Q ± Γ p ) are symmetric and antisymmetric linear combinations. The condition D p = 0 defines the exceptional points, that is, the set of momenta at which H p is not diagonalizable. The momentum dependence of Γ p is crucial for the existence of exceptional points. For Γ p = Γ or Γ p = Γ p+Q , we have Γ a p = 0, and thus D p ≥ ∆ 2 > 0 in the magnetically ordered phase. For Γ a p = 0, the real and imaginary parts of D p yield two conditions for an exceptional point, which need to be satisfied simultaneously. The second condition requires a relaxation rate that exceeds the magnetic gap at particular momenta. The exceptional points can be classified by a topological charge ν i = ± 1 2 via where Γ i is a closed contour encircling the i-th exceptional point counterclockwise [5,39,40]. Exceptional points with opposite charge are connected by branch cuts where D p is real and negative. Fermi surface reconstruction. As an example, we assume a tight-binding dispersion on a square lattice, a magnetic gap ∆ and a wave vector Q = (π − 2πη, π) such that several hole and electron pockets are present, confined by momenta at which the lower and upper quasiparticle bands cross the Fermi level, respectively. The dispersion has the form where t, t , and t are hopping amplitudes between nearest, next-nearest, and third-nearest neighbors, respectively. We use t as our energy unit, and we choose t /t = −0.17 and t /t = 0.05, as widely used for La 2−x Sr x CuO 4 (LSCO) superconductors [29,30,41]. The parameters of the magnetic order ∆/t = 0.144 and η = 0.106 are taken from recent DMFT results for the two-dimensional Hubbard model with LSCO parameters at a hole doping p = 1 − n = 0.177 [30]. For the momentum-dependent relaxation rate Γ p we assume a d-wave form with γ 0 , γ d ≥ 0, which has its minimal value (Γ min = γ 0 ) along the Brillouin zone diagonals, and its maximal value (Γ max = γ 0 + γ d ) at the points (π, 0) and (0, π) on the p x and p y axis. The relaxation rate in a cuprate compound from the LSCO family has recently been determined experimentally via angle-resolved magneto-resistance measurements in the overdoped regime at various temperatures [42], yielding an estimate γ 0 /t ≈ 0.015 and γ d /t ≈ 0.15. For a given relaxation rate Γ p , the condition (7) can always be satisfied for a sufficiently small gap, for instance, near the onset of magnetic order at a quantum critical point. For a better visualization of the topological effects, we choose a sizable magnetic gap and consider relatively large values for the relaxation rate, namely γ 0 /t = 0.05, and γ d /t ≤ 1.6. We fix the doping level at p = 0.177. All results are obtained at zero temperature. In Fig. 1 (top row) we show the "nesting" lines defined by Eq. (6), and the lines corresponding to the condition (7) for different γ d /t. Exceptional points where these lines cross exist for γ d /t ≥ 1.0. Changing parameters, exceptional points can be created or annihilated only in pairs with opposite topological charge. In the bottom row of Fig. 1 we show the quasiparticle Fermi surfaces. Electron and hole pockets are disconnected for γ d /t = 0 and 0.8, while for larger γ d /t they merge at isolated momenta on the branch cuts. The conditions given by Eq. (6) (green "nesting" lines), which separates the region of a p > 0 (gray) and a p < 0 (white), and Eq. (7) (red lines). Exceptional points are situated at the intersection of both lines, and carry the topological charge νi = ± 1 2 . Lower row: The quasiparticle Fermi surfaces defined by Re E ± p = µ for fixed doping p=0.177. Hole pockets (orange) and electron pockets (blue) merge at isolated momenta on the branch cuts, that is, on the parts of the nesting line between exceptional points of opposite charge. In Fig. 2 we show the real and imaginary parts of the quasiparticle bands E ± p for γ d /t = 1 as a function of p x along the upper nesting line in Fig. 1, where the discriminant D p is real (since a p = 0). For |Γ p+Q −Γ p | < 2∆, the square root in Eq. (4) is real, such that E ± p = Re E ± p describes two separate bands. For |Γ p+Q − Γ p | > 2∆, that is, on the branch cut, the square root in Eq. (4) is purely imaginary such that E + p = E − p , while now Γ ± p = −Im E ± p assumes two distinct values. At the exceptional points, where |Γ p+Q − Γ p | = 2∆, both real and imaginary parts of the two complex bands E ± p collapse to a single value. The degenerate band E + p = E − p on the branch cut is dispersive. Thus, it intersects the Fermi level only at isolated momenta, which leads to the peculiar Fermi surface topology in Fig. 1. The merging of hole and electron pockets at single isolated momenta is a generic consequence of exceptional points with opposite topological charge inside the pockets and, thus, not restricted to our specific realization by the particular form of the dispersion in Eq. (9) or the relaxation rate in Eq. (10).
Semiclassical transport through crossing points. In a semiclassical description of the electron dynamics, the momentum of electrons changes smoothly in the direction of the applied force [43]. The Lorentz force acts perpendicularly to the electron velocity, such that a magnetic field makes low-energy electrons move along the Fermi surface. We now clarify how electrons move semiclassically through the crossing points. There are potentially six paths on the Fermi surface. (see Fig. 3). We study the evolution of a biorthonormal basis with left and right eigenstates |L n p and |R n p for the bands n = ± when p traverses the crossing point [44,45]. Since the Hamiltonian in Eq. (3) is symmetric, H p = H tr p , we can choose a gauge such that |L n p = |R n p * . Thus, the Berry connection i L ± p |∂ pα R ± p vanishes and the geometric phase γ B is determined exclusively by the overlap of the ini-+ - Figure 3: Close-up of a crossing point for γ d /t = 1.6. The electron (blue) and hole (orange) pockets encircle exceptional points (red dots) of opposite charge. They merge at one point on the branch cut (red-dashed line). Only the two diagonal paths from Γ1 to Γ3 and from Γ2 to Γ4 are continuously connected. The crossing rules depend on the sign change between the regions a p > 0 (gray) and a p < 0 (white).
tial and final states [46]. For definiteness, the remaining gauge freedom |R n p → ±|R n p has also been fixed. In the Supplemental Material [38] we show that only the two diagonal paths allow for a continuous evolution of the eigenstates through the crossing point. The phase shift is determined by the sign change of a p . With the shorthand notation |n p = |R n p , the transition of states at the crossing point is given by where the transition from left to right is when crossing with sign change + → − (gray to white in Fig. 3) and the evolution from right to left is when crossing with sign change − → + (white to gray in Fig. 3). The first rule in Eq. (11) involves a minus sign beside the well-known swapping of the eigenstates [4]. The velocities ∂ pα E ± p are smooth and finite at the crossing point [38]. Note that the rules in Eq. (11) are gauge dependent, but the total geometric phase accumulated in a closed loop is a gauge independent quantity.
In Fig. 4 we sketch the evolution of the eigenstates for electrons moving along the Fermi surface according to Eq. (11). For the pockets in Fig. 1 we find a vanishing geometric phase γ B = 0 after a completed round, but a relative phase difference π on opposite sides of the hole pocket [step ii) and iv)]. We predict a geometric phase γ B = π for an "eight" topology of a merged electron and hole pocket, so that the original state is then recovered only after two rounds.
Quantum oscillation experiments at sufficiently large magnetic fields ω c τ > 1, where ω c is the cyclotron frequency and τ = 1/2Γ, can be used to measure the Fermi surface topology. Thus, the merging of electron and hole pockets (see Fig. 1) is visible at least in principle in the spectrum of quantum oscillations. A geometric phase can be observed experimentally as a phase shift in quantum oscillations [47]. A detailed microscopic or semiclassical analysis of transport in non-Hermitian systems is still ongoing research [29,[48][49][50][51][52][53][54][55][56] and beyond the scope of this paper.
Spectral functions. The quasiparticle spectral function is given by the diagonal matrixÃ The Green's function in the bare band basis is related to the quasiparticle Green's function by G R p (ω) = U pG R p (ω) U −1 p , where the matrix U p diagonalizes H p for all momenta except, of course, the exceptional ones. The diagonal elements of A p (ω) are obtained as [38] A ↑/↓ p (ω) = 1 2 whereP ± p (ω) are the elements of the diagonal matrix P p (ω) = ReG R p (ω). Due to the momentum shift Q in the spinor Ψ p , the total spectral function for the physical single electron excitations reads For Γ a p = 0 we have Γ D p = 0, so that we recover the wellknown result for a momentum-independent relaxation rate [28,38]. The appearance of the term (14) is directly linked to a nonzero Γ a p . In the Supplemental Material [38] we analyze the effect of exceptional points. Both the second term in (13) and the third term in (14) are discontinuous at the exceptional points. Due to the phase shift π 2 in D p when crossing the exceptional point, the two contributions are mapped onto each other. Thus, the sum of both is continuous. In other words, the nonanalyticity of the complex band at the exceptional points does not appear in A p (ω).
In Fig. 5 we show the spectral function A p (ω) at ω = 0 for γ d /t = 0 and γ d /t = 1. The spectral weight is strongly suppressed for momenta away from the bare Fermi surface [28,38]. Moreover, the angle dependence of Γ p reduces the spectral weight in the antinodal region, such that only Fermi arcs in the nodal region are visible.
Conclusions. We have analyzed the non-Hermitian band topology resulting from a momentum-dependent relaxation rate Γ p in a two-dimensional metal with spiral magnetic order. We provided a concrete example for a specific band dispersion and relaxation rate. We find that arc-shaped branch cuts connecting exceptional points with opposite topological charges appear in the Brillouin zone. Exceptional points inside hole and electron pockets lead to a peculiar Fermi surface topology with pockets merging at isolated points in the Brillouin zone. We have derived rules for the evolution of eigenstates under semiclassical motion through these crossing points, from which geometric phases associated with the Fermi surface topology can be obtained. The change of the Fermi surface topology and the geometric phase are visible at least in principle via quantum oscillations. The spectral function for single-particle excitations, which can be observed in photoemission experiments, exhibits Fermi arcs. Its momentum dependence is however smooth, due to subtle cancellations of the non-analyticities in the complex quasiparticle band structure.
Our work provides an example for an intriguing non-Hermitian topological band structure emerging from a combination of conventional ingredients, in an electron system that was hitherto expected to be topologically trivial. Following this paradigm, we expect the discovery of other condensed matter systems with an interesting non-Hermitian band topology. A: Math. Theor. 47, 035305 (2014).
[45] The right eigenstates are defined by the eigenvalue equation Hp|R n p = E n p |R n p . The left eigenstates are defined by the eigenvalue equation H † p |L n p = (E n p ) * |L n p . Orthonormality is defined by L n p |R n p = δnm, where L n p | = (|L n p ) † . Further properties are discussed in the Supplemental Material [38]. Conductance and Zero Conductance Fluctuation in Non-Supplemental Material for "Non-Hermitian band topology from momentum-dependent relaxation in two-dimensional metals with spiral magnetism" In this Supplemental Material we give further details on (i) the special case of a non-Hermitian part proportional to the identity matrix, (ii) the consequences of off-diagonal self-energy components, (iii) the behavior of the quasiparticle bands and eigenstates when crossing the branch cut, and (iv) the derivation and properties of the spectral functions.

SPECIAL CASE OF A NON-HERMITIAN PART PROPORTIONAL TO THE IDENTITY MATRIX
In this section we briefly discuss the special case where the non-Hermitian part in Eq. (3) is proportional to the identity matrix, which applies to a constant relaxation rate Γ p = Γ and, more generally, to a momentum-dependent relaxation rate obeying Γ p = Γ p+Q . The latter case is fulfilled for a ferromagnetic wave vector Q = (0, 0) and for a Néel antiferromagnetic wave vector Q = (π, π) with a scattering rate of the d-wave form in Eq. (10). The

(non-Hermitian) Hamiltonian reads
where the first term is the Hermitian Hamiltonian H p in Eq. (1). Since the non-Hermitian part is proportional to the identity matrix, it is still possible to diagonalize H p by the same unitary transformation that diagonalizes the Hermitian matrix H p . The eigenvectors of the upper and lower band are with the normalization w ± p = ∆ 2 + a p ± ( a p ) 2 + ∆ 2 2 . We have n p |m p = δ nm for n, m = ±. The diagonalized Hamiltonian readsH p = U −1 p H p U p , which involves the complex eigenvalues The unitary transformation matrix is U p = |+ p |− p . Using the explicit form of the eigenstates in Eq. (S2), one can check that the Berry connection i n p |∂ pα |n p vanishes identically for both bands n = ± and all momenta.
The (diagonal) quasiparticle spectral function matrix is given byÃ components, the quasiparticle spectral functions of the upper and lower bandÃ ± p (ω), are Lorentzians at position Re E ± p − µ = s p ± ( a p ) 2 + ∆ 2 − µ and width −Im E ± p = Γ p . We calculate the spectral function matrix in the band basis † with respect to the quasiparticle spectral functionsÃ ± p (ω). We decompose the quasiparticle Green's function of the upper and lower band intoG R,± p =P ± p (ω) − iπÃ ± p (ω) withP ± p (ω) = ReG ± p (ω) and use G p (ω) = U pGp (ω)U −1 p . Thus, the diagonal elements A ↑ p (ω) and A ↓ p (ω) of the spectral function matrix A p (ω) read One can rewrite the prefactor in front of the quasiparticle spectral functionsÃ ± p (ω) by using the identity The peaks of the quasiparticle spectral functions on the reconstructed Fermi surface defined by Re E ± p = µ are thereby suppressed for momenta away from the bare Fermi surface (defined by p = µ) [S1]. The spectral function for single particle excitations A p (ω) = A ↑ p−Q (ω) + A ↓ p (ω) thus resembles Fermi arcs, since only the pocket surface parts close to the bare Fermi surface are visible. Note that this result is independent of the momentum dependence of the relaxation rate Γ p , which can lead to an additional reduction of the size of the spectral functions.

CONSEQUENCES OF OFF-DIAGONAL SELF-ENERGY COMPONENTS
In the main text, we considered only diagonal non-Hermitian contributions to the Hamiltonian. In the following, we discuss the consequences of non-Hermitian off-diagonal elements. The Hamiltonian H p combined with the most general non-Hermitian part reads with real Γ x p and Γ y p . The corresponding discriminant reads whose zeros define the exceptional points. The diagonal part Γ s p proportional to the identity matrix has no impact on the existence of exceptional points. We assume that we are in the ordered state with ∆ > 0. In the case of Γ a p = 0, a vanishing imaginary part of D p requires Γ x p = 0. Thus, exceptional points do not exist since ( a p ) 2 + ∆ 2 + (Γ y p ) 2 ≥ ∆ 2 > 0. We see that a nonzero Γ a p is necessary for exceptional points. For Γ a p = 0, the existence of exceptional points depends crucially on model and parameter details. We have checked that nonzero Γ x p and Γ y p have only an indirect impact on the quasiparticle dispersions E ± p and the spectral functionsÃ ± p (ω) and A ↑/↓ p (ω) by modifying the value of D p . Thus, we do not expect any major consequence on the conclusions that are presented in the main text.

BEHAVIOR OF THE QUASIPARTICLE BANDS AND EIGENSTATES WHEN CROSSING THE BRANCH CUT
In the main text we have shown that the non-Hermitian Hamiltonian H p in Eq. (3) can have exceptional points at isolated momenta. Exceptional points of opposite topological charge are connected by a one dimensional line, the branch cut, defined by the conditions a p = 0 and (Γ a p ) 2 > ∆ 2 . We now discuss the behavior of the (complex) quasiparticle bands E ± p and the (biorthonormal) eigenstates when crossing the branch cut.
corresponding to Eq. (11) in the main text.
Evolution of the eigenstates and the quasiparticle velocity through the crossing point The previous results do not depend on the precise path in the Brillouin zone through the nesting line, even if the branch cut is crossed. In the main text, we considered a path along the Fermi surface, defined by the constraint Re E ± p = µ. In presence of exceptional points and, thus, branch cuts, hole and electron pockets can merge at a single point, which allows for, in principle, six paths through the crossing point along the Fermi surface. A concrete case is shown in Fig. 3. We now discuss the evolution of the (right) eigenstates |n p ≡ |R n p on the six possible paths. In addition, we show the quasiparticle velocity ∂ pα E n p in α = x, y direction. In Fig. S1, we show the components of |n p and the quasiparticle velocity on the two diagonal paths Γ 1 → Γ 3 and Γ 2 → Γ 4 . The vertical line indicates the momentum at which the hole and electron pockets merge. The respective changes of the eigenstates and bands are indicated. On both paths the branch cut is crossed with a sign change of a p from minus to plus (white to gray in Fig. 3). According to the general result in Eq. (11), there is a transition |+ p → |− p at the crossing point between Γ 1 and Γ 3 , and a transition |− p → −|+ p at the crossing point between Γ 2 and Γ 4 . One can see that the eigenstates and the quasiparticle velocities are continuous at the crossing point, in spite of the branch cut. Hence, these are the paths followed by the electrons in semiclassical dynamics.
In Fig. S2, we show the components of |n p and the quasiparticle velocity on the four non-diagonal paths through the crossing point. For these paths discontinuities are present at the crossing point. finite for almost all momenta. For γ p /t = 1 (right), the nesting line crosses the eight exceptional points (gray vertical lines), at which A ↓,2 p and A ↓,3 p are discontinuous. However, the sum is continuous and does not show any feature at the exceptional points.