Exceptional Topology of Non-Hermitian Systems
Abstract
We review the current understanding of the role of topology in non-Hermitian (NH) systems, and its far-reaching physical consequences observable in a range of dissipative settings. In particular, we elucidate how the paramount and genuinely NH concept of exceptional degeneracies, at which both eigenvalues and eigenvectors coalesce, leads to phenomena drastically distinct from the familiar Hermitian realm. An immediate consequence is the ubiquitous occurrence of nodal NH topological phases with concomitant open Fermi-Seifert surfaces, where conventional band-touching points are replaced by the aforementioned exceptional degeneracies. We furthermore discuss new notions of gapped phases including topological phases in single-band systems, and clarify how a given physical context may affect the symmetry-based topological classification. A unique property of NH systems with relevance beyond the field of topological phases consists in the anomalous relation between bulk- and boundary-physics, stemming from the striking sensitivity of NH matrices to boundary conditions. Unifying several complementary insights recently reported in this context, we put together a clear picture of intriguing phenomena such as the NH bulk-boundary correspondence, and the NH skin effect. Finally, we review applications of NH topology in both classical systems including optical setups with gain and loss, electric circuits, mechanical systems, and genuine quantum systems such as electronic transport settings at material junctions, and dissipative cold-atom setups.
Authors in alphabetic order.
Contents
- I Introduction
- II Non-Hermitian topological band theory
- III Anomalous bulk-boundary correspondence
- IV Physical platforms
- V Concluding remarks
I Introduction
One of the basic axioms of quantum mechanics requires observables, such as the Hamiltonian of a closed system, to be self-adjoint operators, which are typically represented by Hermitian matrices. Real physical systems, however, are at least to some extent coupled to their environment, where the presence of dissipative processes renders their description more complex: In general, the familiar Schrödinger equation with a Hermitian Hamiltonian there is replaced by a Liouvillian super-operator governing the time evolution of the density matrix Breuer and Petruccione (2002). In certain regimes, such open systems in contact with an environment can be accurately described by approaches such as Lindblad quantum master equations Lindblad (1976), Feynman-Vernon theory Feynman and Vernon (1963), and the Keldysh formalism Keldysh (1964). While immensely powerful, the technical complexity of these methods severely limits the range of systems that can be efficiently studied. Effective non-Hermitian (NH) Hamiltonians provide a conceptually simpler and intuitive alternative to fully microscopic approaches, and have already led to profound insights with applications. The spectrum of physical platforms ranges from classical systems, including optical settings, electrical circuits, and mechanical systems, which may be mapped to an effective NH Schrödinger equation, all the way to quantum materials Miri and Alù (2019); Rotter (2009); El-Ganainy et al. (2018); Bender (2007); Ozawa et al. (2019); Datta (2005).
In a wider historical context, effective NH concepts have been ubiquitous for many decades Majorana (1931a); Pancharatnam (1955); Kato (1966); Hatano and Nelson (1996); Berry and O’Dell (1998); Berry (2004); Efetov (1997b); Brouwer et al. (1997); Efetov (1997a); Hatano and Nelson (1997); Silvestrov (1998, 1999), e.g., for describing resonances and broadening in scattering problems in atomic- and particle-physics, as well as in nuclear reactions Majorana (1931a, b); Fano (1961); Breit and Wigner (1936); Feshbach (1958); Feshbach et al. (1954), all the way to applications in biological systems Nelson and Shnerb (1998); Lubensky and Nelson (2000). Following the seminal insight that NH Hamiltonians preserving the combination of parity and time reversal () symmetry stably feature real spectra Bender and Boettcher (1998); Bender (2007), relinquishing the assumption of Hermiticity may even be considered as a fundamental amendment to quantum physics. By now, -symmetric Hamiltonians are well-established as an effective description of dissipative systems with balanced gain and loss El-Ganainy et al. (2018).
In parallel to these developments, the advent of topological phases such as topological insulators and semimetals has revolutionized the classification of matter and led to groundbreaking discoveries of topologically robust physical phenomena Hasan and Kane (2010); Qi and Zhang (2011); Chiu et al. (2016); Armitage et al. (2018). Motivated by experiments reporting novel topological states in dissipative settings Zhou et al. (2018); Bandres et al. (2018); Weimann et al. (2017); Cerjan et al. (2019); Poli et al. (2015); Chen et al. (2017); Hodaei et al. (2017); Zeuner et al. (2015); Helbig et al. (2019), extending the notion of topological phases to NH systems has become a broad frontier of current research. Remarkably, in this context a plethora of uniquely non-Hermitian aspects of topological systems have been revealed Gong et al. (2018); Kunst et al. (2018); Kawabata et al. (2019c); Yao and Wang (2018). Salient examples in the focus of our present review article include an anomalous bulk-boundary correspondence accompanied by the non-Hermitian skin effect Kunst et al. (2018); Yao and Wang (2018); Martinez Alvarez et al. (2018b); Xiong (2018); Lee (2016), the ubiquitous occurrence of exceptional nodal phases Kozii and Fu (2017); Budich et al. (2019); Okugawa and Yokoyama (2019); Zhou et al. (2018); Yoshida et al. (2019b); Szameit et al. (2011); Rui et al. (2019a) with open Fermi-Seifert surfaces Carlström et al. (2019); Carlström and Bergholtz (2018); Lee et al. (2018b), and a new system of generic symmetries Bernard and LeClair (2002) forming the basis for the topological classification of both gapless Budich et al. (2019); Kawabata et al. (2019a) and gapped Esaki et al. (2011); Kawabata et al. (2019c); Lieu (2018b); Zhou and Lee (2019); Shen et al. (2018); Leykam et al. (2017) NH topological phases. In this review, we provide a concise and comprehensive overview of these developments with a special emphasis on their relation to exceptional degeneracies at which both eigenvalues and eigenvectors coalesce, a paramount spectral feature unique to NH systems.
Exceptional degeneracies in NH two-level systems. As an appetizer for the NH Hamiltonian formalism detailed below, we discuss a minimal two-level example that may serve as an intuitive basis for understanding many of the the key concepts unique to NH matrices, in particular the aforementioned exceptional degeneracies. Specifically, we consider the effective Hamiltonian
(1) |
whose complex “energy” eigenvalues
(2) |
generate a generically non-unitary time evolution. Another key observation is that the eigenspectra of NH systems are not analytic in the system parameters, and the diverging response as can be harnessed, e.g., in sensing devices Chen et al. (2017); Hodaei et al. (2017). In contrast to the Hermitian case, the right eigenvectors defined by and left eigenvectors satisfying are generically different. Here, explicitly
(3) |
hence, in clear contrast to Hermitian Hamiltonians, while and are not orthogonal for . Saliently, at the exceptional point (EP), , assumes a Jordan block form, and in addition to the two-fold degeneracy of the eigenvalue at , the eigenvectors coalesce such that only a single right and single left eigenvector remains Heiss (2012) [cf. Eq. (3)]. On a more technical note, at the EP the matrix becomes defective, meaning that the geometric multiplicity (number of linearly independent eigenvectors) is smaller than the algebraic multiplicity (degree of degeneracy in the characteristic polynomial) for the eigenvalue .
To better understand the consequences of this scenario, let us consider tracing a loop in the complex plane with the parameter , so as to enclose the EP at . With we have with on the principal domain. Note that away from the EP, there is always a finite complex energy gap, , and one can thus unambiguously track the energies and the corresponding eigenstates. Remarkably, however, following an eigenstate and its corresponding energy while encircling the exceptional point through one readily finds that
(4) |
This swapping of eigenvalues, being a manifestation of the complex energy living on a two-sheeted Riemann surface known from the behavior of the complex square-root function around the origin, is directly associated with the presence of (second-order) exceptional points [cf. Eq. (2)]. A striking implication is that while encircling the EP at , the real part of the energy crosses zero exactly once, namely when it passes the branch cut on the negative real line, i.e., at . Below, in Section II, precisely this property will be shown to lead to the occurrence of novel NH Fermi arcs, and higher-dimensional generalizations thereof, as a unique and ubiquitous feature of NH band structures.
Outline. The remainder of this review article is organized as follows. In Section II, we discuss in detail the topological band theory of non-Hermitian systems including both nodal phases, which are found to be much more abundant than in the Hermitian realm, and various notions of gapped systems generalizing the concept of insulators. In Section III, we discuss how the bulk-boundary correspondence, i.e., the direct relation between bulk topological invariants and the occurrence of protected surface states, is qualitatively modified in NH systems. This phenomenon is shown to be closely related to the NH skin effect, i.e., the accumulation of a macroscopic number of eigenstates at the boundary of systems with open boundary conditions. In both Section II and Section III, we clarify the direct relation of the uniquely NH phenomenology to the presence or proximity of EPs. In Section IV, we then give an overview of both classical and quantum systems in which the fundamental aspects of NH topology have been predicted to occur or even been experimentally demonstrated already. A concluding discussion is presented in Section V, providing an outlook towards a conclusive understanding of the role of topology in NH systems.
Ii Non-Hermitian topological band theory
In this Section, we systematically review the topological properties of Bloch bands in NH systems. Quite remarkably, the recent pursuit of topologically classifying NH band structures has led to the experimental discovery and theoretical explanation of various topologically stable phenomena that have no direct counterpart in the Hermitian realm, including a novel system of gapped and gapless (symmetry-protected) NH topological phases discussed in this section.
ii.1 Basic concepts and minimal examples
To get an intuitive feeling for the topological properties of NH Bloch bands, we start by discussing some elementary and appetizing examples.
ii.1.1 Topological one-band models
Hatano-Nelson model. Remarkably, and in sharp contrast to Hermitian systems, even a band structure consisting only of a single band may be topologically non-trivial in the NH context. A paradigmatic example along these lines is provided by the Hatano-Nelson model, initially proposed to study localization transitions in superconductors Hatano and Nelson (1996):
(5) |
where () creates (annihilates) a state on site , and with in general [cf. Fig. 1(a)]. The (complex) energy spectrum reads as , and, as a function of , winds around the origin in the (counter)clockwise direction when () as shown in Fig. 1(b). These phases are formally (homotopically) distinguished by the integer quantized value () of the spectral winding number Shen et al. (2018); Gong et al. (2018)
(6) |
A transition between the two topologically distinct regimes then requires for some (here ). We stress the conceptual difference between the topological invariant (6), which distinguishes inequivalent paths in the complex energy plane, and standard Hermitian topological invariants, which quantify some winding of the eigenstates based on the Berry connection. On a more technical note, the Hatano-Nelson model (5) represents a minimal example of a system with a so-called point gap around the singular point in the spectrum Kawabata et al. (2019c) (see Sec. II.3 below for a more detailed discussion). For general multi-band models, we note that is simply replaced by in Eq. (6), where denotes the effective NH Hamiltonian in reciprocal space (Bloch Hamiltonian), such that the winding number in Eq. (6) generically has integer () values.
Non-Hermitian skin effect. The asymmetric hopping strength in the Hatano-Nelson model (5) gives rise to another exotic feature unique to NH systems: In the case of open boundary conditions, a macroscopic number of eigenstates piles up at one of the ends, a phenomenon known as the non-Hermitian skin effect Xiong (2018); Yao and Wang (2018); Martinez Alvarez et al. (2018a); Kunst et al. (2018). The end at which the weight of the eigenstates accumulates depends on which direction of hopping is dominant. This becomes particularly intuitive when one of the hopping directions is entirely turned off, e.g., in Eq. (5). In this case the Hamiltonian with open boundary conditions can be written as a single Jordan block such that the energy spectrum features an exceptional point of order , where is the total number of sites. The proximity to such high-order exceptional points, the order of which scales with system size Martinez Alvarez et al. (2018b); Kunst and Dwivedi (2019), in generic models with open boundary conditions are at the heart of the breakdown of the conventional bulk boundary correspondence as discussed in Section III.
Complex energy vortices. It is natural to consider higher-dimensional extensions of the Hatano-Nelson model. There, we find that zeros in the spectrum lead to the formation of topologically stable vortices in the complex energy. For instance, consider the single-band non-Hermitian nearest-neighbor single-band model corresponding to the spectrum
(7) |
which has vortices (zeros) when both momenta are at or yielding a total of four zeros in the BZ. Focusing on, e.g., the zero at it is clear that it is associated with a finite winding number
A model with a minimal number of two complex zeros can be constructed, e.g., with
(8) |
which nicely displays the stability of the vortices upon varying : The vortices split at a singular point when is increased from zero, and, after traveling in opposite directions through the BZ, merge again at .
On a more conceptual note, the considered two-dimensional systems represent a dimensional extension to a gapless topological phase from a point-gapped lower-dimensional model (the Hatano-Nelson model). This phenomenology bears similarities with Weyl semimetals in the Hermitian realm in 3D that may be seen as families of Chern insulators in 2D, where the Weyl points correspond to topological quantum phase transitions between different Chern numbers Armitage et al. (2018). To see this analogy, we can rewrite Eq. (8) as
(9) |
which, seen as a one-dimensional Hatano-Nelson type model at fixed , changes its winding number [see Eq. (6)] precisely at the position of the complex zeros (vortices). Since the instantaneous 1D model corresponds to unidirectional hopping in the positive direction, and is a simple shift of all energy levels, all eigenstates coalesce being located at the end site in an open chain geometry. Hence, the aforementioned NH skin effect occurs at all , while the winding number changes as a function of , thus highlighting that there is no direct correspondence between these two phenomena unless further assumptions are included as discussed in Sect. III.1.2.
ii.1.2 Two-banded NH models
Since most (topological) properties of NH bands can be analytically understood within the conceptually simple framework of two-banded systems, we proceed by considering NH model Hamiltonians, which in reciprocal space at lattice momentum are of the generic form
(10) |
where with , , the vector of standard Pauli matrices, and is the identity matrix. The complex energy spectrum then explicitly reads as
(11) |
where we dropped the -dependence of all quantities for brevity.
Abundance of exceptional degeneracies. For Hermitian systems (implying ), degeneracies in the spectrum (11) only occur if all three components of are simultaneously tuned to zero. This is the basic reason why topologically stable nodal phases such as Weyl semimetals occur in three spatial dimensions in conventional band structures. However, allowing for in NH systems, from Eq. (11), we see that degeneracies occur when
(12) |
are satisfied simultaneously, i.e., upon satisfying only two real conditions Berry (2004). This implies that nodal points in an NH band structure are generic and stable in two spatial dimensions as shown schematically in Fig. 2.
Another key difference to Hermitian systems is that any non-trivial solutions to Eq. (12) lead to degeneracies in the form of exceptional points, where the NH Hamiltonian becomes defective since the two eigenstates coalesce (become linearly dependent) upon approaching the degenerate eigenvalue. This is not the case for the trivial solution known as a diabolic point. The diabolic point concurs with the aforementioned Hermitian degeneracy condition, but has a much lower abundance as it requires fine-tuning of six parameters in the NH context. These simple algebraic observations on NH matrices have profound implications on the topological classification and physical properties of NH systems, which is elaborated on in the following Section II.2.
ii.2 Nodal phases
A natural question that has recently been the subject of intense theoretical and experimental study is as to what extent the paramount algebraic phenomenon of EPs affects the physical properties of NH systems. In this Section, we review recent results along these lines regarding both the topological classification and physical phenomenology of NH band structures exhibiting EPs.
ii.2.1 Topological non-Hermitian metals
We illustrate the stable occurrence of NH nodal points in two spatial dimensions (2D) by perturbing a Hermitian 2D Weyl point described by the model Hamiltonian
(13) |
in an NH fashion in various ways. The Hermitian perturbation is readily seen to immediately open a gap of order [see Eq. (11) and Fig. 3], demonstrating the fine-tuned character of a 2D Weyl point in the Hermitian realm. By contrast, if we add the corresponding anti-Hermitian perturbation , from plugging into Eq. (12), we find a ring of exceptional degeneracies at , i.e., the system remains gapless (cf. Fig. 3). However, when considering the combination of these two perturbations, Eq. (12) amounts to , meaning that there is a gap as soon as both and are finite, thus rendering the aforementioned nodal ring unstable. More precisely, in Section II.2.2, we will discuss that such nodal structures of higher dimensions are stable only in the presence of certain NH symmetries.
Next, we choose an anti-Hermitian term , which at gives rise to degeneracies when and , i.e., at the isolated points (Fig. 3). By contrast to the ring degeneracy, these isolated EPs are stable against , and for that matter any small NH perturbation. More specifically, the isolated EPs will continuously move in momentum space as a function of generic perturbations, and can only be removed if they meet in momentum space. This renders NH 2D systems with isolated nodal points in the form of EPs a topologically stable phenomenon defining an NH Weyl phase. On a more formal note, as already mentioned in Section I, the complex energy spectrum at an isolated second-order EP behaves like a complex square-root function around the origin. Hence, such EPs in 2D form branch points in energy that can only be removed by contracting the branch cut connecting them.
An important physical consequence of the concomitant phase winding of the complex energy around the EPs is the existence of contours with purely imaginary (purely real) energy emanating from them, also called NH Fermi arcs (imaginary NH Fermi arcs), which are equivalent to the aforementioned branch cuts [cf. Fig. 2(b)]Kozii and Fu (2017); Zhou et al. (2018); Carlström and Bergholtz (2018); Yang et al. (2019b). While in our simple continuum model such contours can extend to infinite momenta, the compact nature of reciprocal space (first Brillouin zone) in Bloch bands describing crystalline structures strictly enforces them to form open arcs connecting the EPs, somewhat reminiscent of Fermi arcs in conventional 3D semimetals. However, a crucial difference to this Hermitian counterpart is that NH Fermi arcs are a bulk phenomenon (similar in this regard to standard Fermi surfaces), while the surface Fermi arcs in 3D Weyl semimetals connect the projection of the Weyl points to a given surface Armitage et al. (2018). Thus, 2D NH Weyl phases distinguished by the number of pairs of EPs are an NH counterpart of metallic dispersions in solids, while generic Hermitian 3D Weyl systems represent semimetallic band structures in this solid state context.
Knotted non-Hermitian metals. Moving to three spatial dimensions (3D), the simple parameter counting in Section II.1.2 tells us that EPs in 3D generically, i.e., without relying on fine-tuning or symmetries, form closed nodal lines in reciprocal space rather than occurring at isolated points Xu et al. (2017); Cerjan et al. (2019). This allows for a new category of topologically stable NH metallic phases where the nodal lines themselves represent topologically non-trivial objects such as links Carlström and Bergholtz (2018); Yang and Hu (2019) or knots Carlström et al. (2019); Stålhammar et al. (2019). By slicing such 3D systems into layers of 2D systems in reciprocal space, the aforementioned argument on NH Fermi arcs may be readily generalized to the 3D case in the following sense: Exceptional nodal lines necessarily bound open NH Fermi surfaces, which for knotted nodal structures appear in the form of Seifert surfaces (cf. Fig. 4). Quite remarkably, these phenomena are not just mathematical possibilities of academic interest, but in fact quite simple microscopic tight-binding models within experimental reach have recently been shown to exhibit a rich variety of linked and knotted NH nodal structures Carlström et al. (2019); Stålhammar et al. (2019); Li et al. (2019a); Yang et al. (2019a). The phenomenon of knotted nodal NH band structures has no direct counterpart in Hermitian systems. There, due to the higher co-dimension of nodal points additional symmetries are necessary to stabilize (knotted or linked) nodal lines Bi et al. (2017), and such fine-tuned nodal structures would not entail Fermi-Seifert surfaces.
ii.2.2 Symmetry-protected nodal phases
Requiring symmetries is well known to generally refine a topological classification by constraining the set of eligible physical systems. Concretely, two model Hamiltonians that would be considered equivalent in absence of a given symmetry may become distinct in its presence, if any path adiabatically connecting them necessarily breaks that symmetry. This phenomenon defines the notion of symmetry-protected topological (SPT) phases Chen et al. (2013); Chiu et al. (2016).
Symmetries in Hermitian systems. In conventional Hermitian systems, a primary example of nodal SPT phases is provided by Dirac semimetals. There, the spin-degenerate Dirac points may be continuously removed individually, unless protecting symmetries such as the combination of parity and time-reversal symmetry (TRS) are postulated. This is in contrast to Weyl semimetals, the individual Weyl points of which are topologically stable without symmetries other than the lattice momentum conservation defining the Bloch band structure.
A first comprehensive symmetry classification was achieved in a seminal paper by Altland and Zirnbauer (AZ) Altland and Zirnbauer (1997). The AZ classification is based on generic symmetry constraints characterizing ensembles of mesoscopic systems beyond standard unitary symmetries that commute with the system Hamiltonian. Specifically, the considered constraints are the anti-unitary TRS defined by
(14) |
where the asterisk denotes complex conjugation, the particle-hole constraint (PHC)
(15) |
and, resulting from the combination of TRS and PHC, the chiral symmetry (CS)
(16) |
Considering all independent combinations of these constraints gives rise to the ten AZ symmetry classes, on the basis of which the celebrated periodic table of topological insulators has been constructed Schnyder et al. (2008); Kitaev et al. (2009); Ryu et al. (2010). Later on, also considering conventional (commuting unitary) symmetries such as crystalline symmetries has resulted in the identification of a plethora of additional – both gapped and nodal – topological band structures Fu (2011); Ando and Fu (2015); Chiu et al. (2016).
Generic symmetries in non-Hermitian systems. The natural question of how the AZ symmetry classification may be generalized to NH systems was addressed by Bernard and LeClair (BLC) Bernard and LeClair (2002), who derived a 43-fold symmetry classification for ensembles of NH matrices. This system of symmetries was proposed for the topological classification of bosonic Bogoliubov-de Gennes Hamiltonians by Lieu, 2018b. Here, we would like to briefly review key elements of the general BLC classification and its recently proposed amendments Kawabata et al. (2019c), focusing on qualitative differences to the AZ classification in Hermitian systems. In essence, the main complication in NH systems compared to the Hermitian realm is that transposition () and complex conjugation () are inequivalent operations, and even Hermitian conjugation () may act non-trivially on a given NH effective Hamiltonian [see Kawabata et al. (2019c) for a detailed discussion along these lines]. As a consequence, both TRS and PHC split into two inequivalent NH generalizations, distinguished by whether complex conjugation is replaced by transposition or not in Eq. (14) and Eq. (15), respectively. Furthermore, the non-trivial action of Hermitian conjugation gives rise to so-called pseudo-Hermiticity constraints Mostafazadeh (2002)
(17) |
where () would be ordinary commuting (chiral anti-commuting) symmetry constraints for Hermitian , but give rise to new symmetry classes in the generic NH case. Note that symmetries involving Hermitian conjugation leave the (quasi)momentum invariant, and thus lead to local constraints in reciprocal space, which change the codimension of the EPs in the complex spectra of Bloch Hamiltonians (as discussed below).
Since an additional minus sign upon complex conjugation may be generated simply by multiplication with the imaginary unit (), TRS and PHS as defined in Eq. (14) and Eq. (15) may be mapped to one another by considering instead of Kawabata et al. (2019b). The identification of these operations for the sake of classification then is, at least at a formal level, justified by the fact that the spaces of eligible Hamiltonians differing by a prefactor of are clearly isomorphic. However, since physically a multiplication by has quite dramatic effects, it is fair to say that in real models these two cases may still correspond to quite different scenarios (see also Section II.4). Taking into accounts all aforementioned symmetry constraints and relations, a careful counting of all independent symmetry classes leads to a grand total of 38 Kawabata et al. (2019c), rather than the 43 symmetry classes originally proposed by BLC.
NH symmetries and abundance of EPs. Given this NH symmetry classification, we would now like to review and illustrate the drastic effect of NH symmetries on the occurrence and stability of exceptional nodal structures in NH band structures Budich et al. (2019). As discussed in Section II.1.2, in the absence of symmetries EPs have codimension two [see Eq. (12)], and thus generically appear at isolated points in 2D and closed lines in 3D NH band structures.
Some basic intuition about how NH symmetries change this behavior can be gained again by considering two-banded models as introduced in Sec. II.1.2 preserving the symmetry . For concreteness, we make the explicit choice . Then, the symmetry (17) in Eq. (10) implies the constraint which trivializes one of the conditions, namely [see Eq. (12)], for obtaining EPs. Thus, the codimension of exceptional degeneracies is reduced from to . As an immediate consequence, EPs at isolated points appear in 1D, and closed lines of EPs occur in 2D (see, e.g., Fig. 3 for the appearance of an exceptional ring). This dimensional shift promotes the aforementioned bulk Fermi arcs to open Fermi volumes, as the surfaces bounded by the EPs now have the same spatial dimension as the system itself.
This phenomenology is not limited to the minimal two-band setting at hand but has been shown to generalize to generic NH band structures in numerous BLC classes that contain reality constraints on the complex spectrum Budich et al. (2019). A -theory based classification of gapless nodal NH phases has very recently been reported by Kawabata et al., 2019a thus arriving at periodic tables encompassing all symmetry classes proposed by Kawabata et al., 2019c. Instructive examples starting from four-band Dirac models have been worked out explicitly by Rui et al., 2019b and symmetry protected rings of EPs are know to naturally emerge in honeycomb based systems Yoshida et al. (2019b); Szameit et al. (2011).
ii.3 Gapped phases
We now turn to the topological classification of gapped NH systems, again focusing on crucial differences to the conventional Hermitian realm, where the periodic table of topological insulators and superconductors based on the AZ symmetry classification by now has become a widely known amendment to the theory of Bloch bands.
ii.3.1 Point gaps vs. line gaps
The first crucial observation when moving to NH band structures with complex energy spectra is that there is no canonical way of defining a spectral gap. To overcome this issue, Kawabata et al., 2019c recently proposed to classify complex-energy gaps into two categories: point and line gaps. An NH model is said to have a point gap when the complex energy bands do not cross a base point , and where crossing this base point defines a gap closing transition (cf. Fig. 5). A line gap, on the other hand, is defined by a line in the complex energy plane, which has no intersections with the energy bands (cf. Fig. 5). Note that models with a line gap also always have a point gap. Line gaps in complex spectra carry close similarities to energy gaps in Hermitian models Kawabata et al. (2019c), as a spectrum of a Hermitian model is said to be gapped when there are no energy bands that cross the Fermi energy . Indeed, in both the Hermitian and NH case, the individual bands in a spectrum with a line gap can be contracted to single points. Point gaps that do not generalize to line gaps do not have a direct Hermitian counterpart and are thus genuinely non-Hermitian.
Recently, it was shown that -dimensional NH models with a point gap can be interpreted as the “surface theory” of -dimensional, Hermitian models Lee et al. (2019d); Gong et al. (2018); Foa Torres (2019). Following Lee et al., 2019d, this relation may be intuitively understood at the level of the Hatano-Nelson model [cf. Eq. (5)]: In the long-time limit, there is only one chiral mode in the system. Indeed, at , i.e., , it is possible to find two modes with opposite chirality: One mode with group velocity , and another mode with . The lifetime of these modes is set by , which is found to be positive (negative) for the mode with group velocity (). In the long-time limit, only the mode with survives, such that we are left with a single chiral mode. In this sense, this one-dimensional non-Hermitian model realizes the anomalous edge behavior of the two-dimensional quantum Hall effect, and can thus be interpreted as the “edge theory” of the latter Lee et al. (2019d).
ii.3.2 Symmetry-protected point-gapped phases
The base energy with respect to which a point gap is defined may without loss of generality be chosen as , which at most amounts to adding a constant complex energy shift to a given Hamiltonian. Then the set of all admissible NH Bloch Hamiltonians is simply given by the general linear group formed of all regular complex matrices , where is the number of bands. Without additional symmetries, the set of inequivalent (strong) topological NH phases in spatial dimensions for is then given by
(18) |
i.e., by the -th homotopy group of Schnyder et al. (2008); Budich and Trauzettel (2013). Remarkably, non-symmetry protected topological NH band structures thus occur in odd spatial dimensions Gong et al. (2018), in stark contrast to Hermitian systems, where the -th Chern number in characterizes topological band structures that do not rely on additional symmetries Ryu et al. (2010). For the simplest conceivable case , the explicit invariant characterizing a given model Hamiltonian is given by the spectral winding number defined in Eq. (6). This can be generalized to arbitrary by simply replacing , and to odd as a standard higher-dimensional analog of the winding number known from chiral symmetric systems in the Hermitian realm [see Eq. (20) below] Ryu et al. (2010). This correspondence is not a coincidence, and it has been shown by Gong et al., 2018 that any NH Hamiltonian may be augmented to a CS preserving Hermitian Hamiltonian
(19) |
acting on a doubled Hilbert space, such that the standard Hermitian chiral invariant associated with concurs with the NH spectral winding invariant
(20) | |||||
in dimensions Schnyder et al. (2008); Budich and Trauzettel (2013). Based on these observations and the AZ symmetry classification Altland and Zirnbauer (1997) (see also Section II.2.2), Gong et al., 2018 arrived at a first NH counterpart of the periodic table of topological insulators. However, as discussed in more detail in Section II.2.2, the non-trivial action of Hermitian conjugation in NH systems naturally refines the 10-fold AZ classification to the 43-fold BLC classes, later proposed to be reducible to a 38-fold way Kawabata et al. (2019c). Adapting the -theory methods used by Kitaev Kitaev et al. (2009) for the Hermitian periodic table to this NH scenario, topological classification tables for gapped phases based on the BLC symmetry classification have recently been derived Kawabata et al. (2019c); Zhou and Lee (2019).
ii.3.3 Symmetry-protected line-gapped phases
Regarding gaps in the shape of a straight line in the energy spectrum, in principle any offset and orientation in the complex plane may be considered to start with. However, similar to the point-gap case, by means of a constant energy shift, the gap line may be transformed to cross the origin. Furthermore, by rescaling the Hamiltonian with a complex constant, such a gap line may then be rotated to, say, the real energy axis. Since such a rotation of the energy spectrum may violate, or at least transform, generic NH symmetries, the authors of Ref. Kawabata et al., 2019c still distinguished line-gaps along the real and imaginary axis, due to their distinct behavior under spectral (reality) constraints.
For the case of a real line gap, any NH model Hamiltonian may be continuously deformed into a Hermitian Hamiltonian without breaking of symmetries, which reduces the classification problem to that of Hermitian matrices. In the case of an imaginary gap a similar deformation to an anti-Hermitian Hamiltonian is always possible. However, since the NH symmetries may be transformed in a non-trivial way when rotating to the Hermitian Hamiltonian , the classification problem of NH Hamiltonians with imaginary line gaps amounts to that of Hermitian systems up to a shift in symmetry class. Based on these observations, periodic tables for line-gapped Hamiltonians in all symmetry classes have been obtained by Kawabata et al., 2019c. Furthermore, Liu and Chen, 2019 considered the classification of defects in the BLC classes and generalizations thereof.
ii.4 Complementary classification approaches
So far, our discussion of NH topological band structures has been based on the BLC symmetry classification, which is a direct NH generalization of the celebrated AZ classification of electronic systems in the Hermitian realm. Given the broad spectrum of applications of effective NH Hamiltonians (see Section IV for an overview), depending on the given physical situation, differing from the BLC classification by considering other symmetries and physical constraints can be natural. In the following, we briefly highlight some prominent examples of deviations from the classification discussed in Sections II.2-II.3.
ii.4.1 Other symmetries
The combination of time-reversal symmetry and parity, widely known as -symmetry, was originally considered as a fundamental NH amendment to quantum physics Bender and Boettcher (1998), as it gives rise to reality constraints on the spectrum known as pseudo-Hermiticity Mostafazadeh (2002), similar to the aforementioned constraint [see Eq. (17)] from the BLC system of symmetries. By now, -symmetry is widely established in effective NH descriptions of a variety of physical settings including photonic systems Regensburger et al. (2012); Özdemir et al. (2019); Feng et al. (2017); Zyablovsky et al. (2014); Yuce (2015). In particular, in the context of NH topology, states protected by -symmetry had been observed in optical systems Weimann et al. (2017) even before the systematic classification of NH symmetry-protected topological phases (see Sections II.2-II.3) was reported. As a second example outside of the BLC classification, a loose analog of supersymmetry, considered in high energy as a fundamental amendment to the standard model, has been identified in certain optical settings Miri et al. (2013); Heinrich et al. (2015). Moreover, Liu et al., 2019a have classified NH phases with reflection symmetry.
ii.4.2 Fundamental constraints in quantum many-body systems
The BLC symmetry classification applies to generic NH matrices. However, when NH Hamiltonians are employed to effectively describe some form of dissipation in quantum many-body systems, inherent physical constraints reduce the space of eligible matrices. For example, the spectrum of effective Hamiltonians derived from a retarded Green’s function including an NH self-energy is constrained to lie in the lower complex half-plane (see Bergholtz and Budich, 2019 for a recent discussion in the context of NH topological phases). This immediately rules out spectral winding around the origin [cf. Eq. (6)] and vortices in the complex spectrum as discussed in Section II.1.1, thus directly affecting the topological classification. A similar constraint appears when considering Liouvillian operators governing the dynamics of open quantum systems as NH matrices Song et al. (2019a); Lieu et al. (2019). The basic physical meaning of such constraints is that quantum dissipation can damp out energy eigenstates (negative imaginary part) or leave them decoherence free (zero imaginary part), but not amplify their weight, which would correspond to a positive imaginary part.
ii.4.3 Homotopy perspective
Finally, we note that from Hermitian systems it is well known that there are so-called fragile topological phases (for a recent discussion see, e.g., Kennedy, 2016) that do not survive the addition of extra bands. Such phases are not captured by the K-theory approach of the classification schemes described above, but can be described within a homotopy-theory based classification Kennedy and Zirnbauer (2016); Kennedy (2016). In the NH context, new fragile topological phases have recently been uncovered by analyzing NH band structures from the vantage point of homotopy Li and Mong (2019); Wojcik et al. (2019). It is worth noting that such fragile phases relying on a low number of bands even exist in the absence of additional symmetries.
Iii Anomalous bulk-boundary correspondence
In this Section, we review recent findings on a striking phenomenology unique to NH systems, namely qualitative changes in the so called bulk-boundary correspondence (BBC) – a fundamental principle for topological phases Hasan and Kane (2010). In conventional Hermitian systems, the BBC establishes a one-to-one relation between topological invariants defined for infinite periodic systems, and protected gapless boundary states occurring in systems with open boundaries. By contrast, in NH topological systems, the BBC in its familiar form is found to generically break down (see Section III.1), and qualitative amendments to re-establish a modified NH BBC have been proposed (see Section III.2). For clarity, the conventional BBC known from Hermitian systems is in the following referred to as cBBC.
iii.1 Breakdown of the conventional bulk-boundary correspondence
In this Section, we review the mechanisms that lead to the breakdown of cBBC, i.e., the failure of topological invariants computed from the Bloch Hamiltonian to correctly predict the existence of boundary states. Furthermore, we discuss the NH skin effect as well as the spectral instability of NH matrices, which accompanies the breakdown of cBBC. There is, however, no strict one-to-one relation with these phenomena as the skin effect can occur also in systems where the cBBC does not have a clear meaning as, e.g., in systems with only point gaps.
iii.1.1 Canonical models and their interrelation
The breakdown of cBBC in NH models was first observed by Lee, 2016, where a Creutz ladder with complex hopping terms and onsite dissipation [see bottom panel of Fig. 6(a)] was studied. This phenomenon may be attributed to the anomalous behavior of the bulk states that, in the case of open boundary conditions (OBC), pile up at the boundaries Xiong (2018); Kunst et al. (2018); Xiong et al. (2016) (see also Section III.1.2 for a more detailed discussion). The easiest, and most intuitive way of breaking cBBC is by including hopping terms in the tight-binding Hamiltonian, whose tunneling strengths are direction dependent (anisotropic) [see upper panel of Fig. 6(a)]. As a consequence, the bulk states can propagate around the system in the preferred direction for periodic boundary conditions (PBC), while they are found to pile up at the boundaries in the case of OBC [see Fig. 6(b)]. This extreme difference in the behavior of the bulk states under different boundary conditions quite intuitively invalidates the authority of bulk topological invariants computed for PBC in determining the existence of boundary states. To explicate and exemplify this exotic behavior, we start by studying a one-dimensional, conceptually simple example, which displays similar features as reported by Lee, 2016. We consider an NH version Lieu (2018a) of the Su-Schrieffer-Heeger (SSH) chain Su et al. (1980) as described by the Hamiltonian
(21) | |||
where () creates (annihilates) a state on sublattice site in unit cell , is the total number of unit cells, and are the nearest-neighbor (nn) hopping parameters inside the unit cell, and is the nn hopping parameter between unit cells [see top panel of Fig. 6(a)] Kunst et al. (2018); Yin et al. (2018); Yao and Wang (2018). Hermiticity is broken when , which results in a different magnitude of the hopping amplitude between hopping to the left with respect to hopping to the right inside the unit cell. The Bloch Hamiltonian is of the general form given in Eq. (10), here with
Similar to the Hermitian SSH chain, this model has a chiral symmetry, i.e., , and it is thus possible to define a winding number, where in the Hermitian case, this winding number determines the number of states localized to the ends Ryu and Hatsugai (2002); Schnyder et al. (2008). The nonzero values of the NH generalization of the winding number Gong et al. (2018); Kawabata et al. (2019c), i.e., the spectral winding number [cf. Eq. (6) with replaced by ], are indicated explicitly in the spectrum in the inset in Fig. 6(b) by the green shaded areas. Here, as opposed to the conventional case, the winding number spectacularly fails to predict the existence of the end states in the OBC case, which are shown in red in the OBC spectrum. In fact, the winding number changes value when a gap closing appears in the PBC spectrum, which is at strikingly different parameter values compared to when the OBC system features phase transitions. To further elucidate what is going on we plot the sum of the amplitude per site for all wave functions in the case of OBC in Fig. 6(b), confirming that the wave functions indeed pile up at the boundary. In summary, the simple model defined by Eq. (21) indeed breaks cBBC and exhibits NH-skin-effect behavior, which was very recently confirmed in several experiments Ghatak et al. (2019); Helbig et al. (2019); Hofmann et al. (2019b); Xiao et al. (2019).
We note that models in which the modulus of the hopping amplitudes is explicitly direction-dependent, such as in Eq. (21), are sometimes referred to as “nonreciprocal hopping models” in the literature Hofmann et al. (2019b), and the accumulation of bulk states at a boundary is often attributed to this property Lee et al. (2019a). While this bears analogy with nonreciprocal optical models, where the symmetry of wave transmission is broken (see Sounas and Alù, 2017 for a review), the straightforward translation of this definition to the language of tight-binding models with internal degrees of freedom may not be unambiguous. On a quite obvious note, when interpreting the sublattice degree of freedom of the NH SSH model in Eq. (21) as a spin rather than a spatial degree of freedom, the model would no longer be nonreciprocal in the aforementioned sense: While the internal coupling strengths between the two spins have a different magnitude, the hopping magnitude between lattice sites is no longer direction dependent. Nevertheless, in this different interpretation, the model still exhibits all the aforementioned properties. We now demonstrate that the ambiguity of this notion of reciprocity goes even much further.
In particular, a simple unitary transformation relates the Bloch Hamiltonians of the NH SSH model and the Lee model:
(22) |
Here we can directly identify , for Lee (2016). In this model it is natural to interpret as onsite gain () and loss (), while and remain standard Hermitian nn hopping parameters inside and between unit cells, respectively [cf. the lower panel in Fig. 6(a)]. Moreover, also with OBC, it is easy to show that one may write , where is the identity matrix of dimension , such that with again being unitary. Thus, the spectra of Lee’s model Lee (2016) with either PBC or OBC are identical to those of the NH SSH model as shown in the inset in Fig. 6(b). It also follows that the Lee model also exhibits a similar accumulation of bulk states at the boundary, since the unitary transformation only acts locally and hence does not drastically alter the localization of the eigenstates. ,
In summary, while Lee’s model [see bottom panel of Fig. 6(a)] only contains diagonal onsite gain and loss terms, it is related to a model with explicitly anisotropic hoppings through a local unitary transformation. This observation further blurs the difference between “reciprocal” and “nonreciprocal” tight-binding models as inferred from the symmetries of their hoppings, and we thus refrain from such a distinction in this review. Instead, we emphasize that the breakdown of the cBBC is a generic NH phenomena fundamentally independent of the microscopic provenance of the non-Hermiticity.
iii.1.2 Non-Hermitian skin effect
The concept of a BBC relies on the doctrine that introducing boundaries into a model does not have significant effects on the bulk states meaning that the model does not undergo a topological phase transition when going from PBC to OBC. In stark contrast, the behavior of the bulk states associated with the family of cBBC-breaking NH models studied in this Section is altered in an extreme way upon considering OBC: These models feature the NH skin effect [cf. Fig. 6(b)], a term coined by Yao and Wang, 2018.
Intuitively, the appearance of the localized bulk states, which are also called skin states, can be understood from the presence of or proximity to one or more high-order EPs through which the states need to pass when tuning from PBC to OBC Xiong (2018). The appearance of these EPs, which scale with system size (infinite order EPs occur in the thermodynamic limit) similar to what we saw for the Hatano-Nelson model in Sect. II.1.1, results in a topological distinction between the model with PBC and OBC thus leading to a natural breaking of cBBC Xiong (2018). The connection between higher-order EPs, say th-order EPs at which eigenstates coalesce (cf. Sect. II.2), and the piling up of bulk states can then be understood as follows: Close to such an EP, a macroscopic number of eigenstates necessarily have large (spatial) overlap, which is achieved through their accumulation at the same boundary.
Importantly, this NH skin effect always appears when cBBC is broken, and can thus be seen as a telltale signature thereof. The anomalous localization behavior of the bulk states does not find a counterpart in Hermitian physics and is thus an inherently NH phenomena.
A natural question to ask is what minimal ingredients are needed for an NH hopping model to possess skin states, and thereby to break cBBC. While not being a sufficient criterion, a necessary requirement is that the Hermitian, , and anti-Hermitian, , parts of the NH Hamiltonian, , do not commute, i.e., .^{1}^{1}1An equivalent way of stating the necessary condition for skin states is that cannot be normal. If they would commute, then and share a common eigenbasis, which means that the eigenstates of are the eigenstates of a Hermitian matrix, namely, of (and ), and as a consequence, the corresponding eigenstates form a standard orthonormal basis and can as such not be skin states.
In Longhi, 2019c, it is shown that the existence of the NH skin effect in one-dimensional NH models can be detected by making use of a bulk probe: If the maximum value of the Lyapunov exponent in the long-time limit is at a drift velocity different from zero, this is a sufficient condition for the NH model to display the NH skin effect as well as symmetry-breaking phase transitions in the OBC spectrum.
A very recent suggestion is that the presence of a topologically nontrivial point gap in the complex energy spectrum of the Bloch Hamiltonian is equivalent to that the eigenstates in the OBC system are skin states Zhang et al. (2019a); Okuma et al. (2019), which was also observed in Wanjura et al., 2019 in the context of driven cavity arrays with dissipation. While this observation is tempting, its precise domain of validity remains to be explored.
One may ask whether the piling up of states is forbidden by certain symmetries. Indeed, by Kunst and Dwivedi, 2019 it is shown that -symmetric models (cf. Sect. II.4) in the -unbroken phase Bender and Boettcher (1998) cannot possess skin states, which is corroborated in Kawabata et al., 2019c, where it is additionally shown that models in the presence of a parity (inversion) symmetry, TRS, which is defined as the relation in Eq. (14) with , or pseudo-Hermiticity in the unbroken phase (cf. Sect. II.2.2) are also excluded from exhibiting a breakdown of cBBC. It can be intuitively understood why these symmetries prevent the existence of skin states: For example, symmetry maps one boundary to the opposite boundary, such that any state localized at only one of the boundaries automatically breaks Hu and Hughes (2011).
It is worthwhile pointing out that skin states do not necessarily have to accumulate on one boundary alone Song et al. (2019b). For example, taking two time-reversed copies of a cBBC-breaking model immediately results in the appearance of skin states on both boundaries, which is referred to as the skin effect by Okuma et al., 2019. Additionally, skin states may also appear on boundaries with a co-dimension higher than one, such as at corners and hinges Edvardsson et al. (2019a); Ezawa (2019d); Luo and Zhang (2019); Liu et al. (2019b).
iii.1.3 Spectral instability
Similar to the eigenstates, also the eigenvalues of cBBC-breaking NH Hamiltonians are extremely sensitive to perturbations that connect the boundaries [see, e.g., the inset in Fig. 6(b)]. This sensitivity to boundary conditions can even result in drastically different qualitative features of the two spectra: Indeed, the OBC spectrum may for instance be gapped and topologically non-trivial, while the PBC spectrum of the same model is gapless Kunst et al. (2018). This spectral instability can be systematically understood as a discontinuous behavior in the eigenvalue spectra of NH matrices under small random perturbations. More specifically, while adding a small perturbation with largest absolute eigenvalue to a Hermitian systems at most leads to a change of order in the spectrum, in NH matrices changes of order may occur Krause (1994), where is the number of sites. In the thermodynamic limit, , this amounts to a change of order one for an arbitrarily small representing the analytical reason for the observed fragility of eigenvalue spectra in NH systems. The tuning between boundary conditions may be interpreted as such a perturbation (see Herviou et al., 2019a for a detailed discussion).
This spectral instability is intimately related to the previously discussed NH skin effect: A study of the spectral instability in a cBBC-breaking model revealed that when tuning between OBC and PBC, one or more higher-order EPs are encountered Xiong (2018). Indeed, when the boundaries of an NH model with OBC are connected via an exponentially small perturbation proportional to the system size , i.e., for some model-dependent constant Kunst et al. (2018); Koch and Budich (2019), the spectrum shows crossover behavior, which can be understood from the behavior of the skin states: For a large enough coupling, which is found to be exponentially small in , the skin states can tunnel through and behave like ordinary bulk states in the sense that they are evenly distributed throughout the lattice in which case the spectrum qualitatively resembles that of the PBC case Kunst et al. (2018). Additionally, the presence of perturbations connecting the boundaries was shown to result in unconventional behavior for the fidelity and Loschmidt echo near the higher-order EPs Longhi (2019a).
Due to the extreme sensitivity of cBBC-breaking NH models to boundary conditions, it seems natural to wonder about the physical relevance of studying the eigenvalue spectra of such NH models with OBC Gong et al. (2018); Herviou et al. (2019a). However, when requiring physically-motivated locality conditions on the considered perturbations, the physical properties specific to the eigenspectra of NH systems with OBC have been shown to be robust Koch and Budich (2019). This renders the anomalous BBC observed in the eigenvalue spectra of NH systems a topologically stable and generically observable phenomenon.
iii.1.4 Domain wall geometries
So far, we have focused on the physics of cBBC-breaking NH models in the case of PBC and OBC, and noticed that both the quantitative and qualitative behavior of these NH models can be extremely different in these two cases. Another interesting geometry to consider is that of domain walls, which can lead to drastic alterations of the physics of NH models Schomerus (2013); Malzard et al. (2015); Malzard and Schomerus (2018). For example, Xiong, 2018 pointed out that if a cBBC NH model is coupled to another (NH) model that resides in a different topological phase, high-order EPs disappear rapidly from the spectrum. It has been conjectured Xiong (2018); Leykam et al. (2017) that cBBC is generically restored in such domain wall geometries. However, Kunst et al., 2018 explicitly exemplified that upon coupling the NH SSH model [cf. Eq. (21)] to its Hermitian, topologically trivial counterpart, cBBC may remain broken in the sense that the skin effect prevails. Indeed, the proximity to EPs persists, and bulk states still locally accumulate, albeit now at the domain wall, as long as the energy gap in the Hermitian system is large enough Kunst et al. (2018) (see Fig. 7). For a sufficiently small gap (or short Hermitian domain), the skin states can tunnel through, and behavior similar to the NH model with PBC is retrieved Kunst et al. (2018); Herviou et al. (2019a). Changing the size of the band gap in the attached Hamiltonian in such a setup may thus be seen as an alternative way of tuning between OBC and PBC, while at the same time introducing new effects that cannot be observed in simple OBC geometries.
Intriguing domain wall effects have also been studied for models that preserve cBBC. For example, by coupling two -symmetric SSH chains, which are in distinct topological phases, a defect state appears at the domain wall with positive imaginary energy, thus representing a solution with a growing amplitude, while the bulk states all have zero imaginary energy Schomerus (2013). A very similar setup was also considered by Yuce, 2018, confirming that PT symmetry is indeed spontaneously broken on the interface. As a consequence, the defect state dominates in the long-time limit. These predictions were indeed experimentally confirmed in a resonator chain Poli et al. (2015) paving the way to the experimental realization of topological lasers St-Jean et al. (2017); Zhao et al. (2018); Parto et al. (2018). Additionally, it has been shown that models which are topologically trivial in the Hermitian limit can host topologically protected defect states in the NH case Malzard et al. (2015); Malzard and Schomerus (2018). Considering domain walls in the form of defects can thus lead to new genuinely NH physical phenomena.
iii.2 Approaches to re-establishing the bulk boundary correspondence in NH systems
While the concept of a BBC in NH models with a point gap in their complex spectra has largely remained elusive, significant progress has been made on re-establishing an NH BBC in models with line gaps Kunst et al. (2018); Yao and Wang (2018); Yao et al. (2018); Xiong (2018), which will be the focus of our subsequent discussion. We will review two main approaches in detail: (i) The biorthogonal BBC approach, which makes direct use of the properties of the OBC spectrum and relates phase transitions to delocalization transitions of biorthogonal boundary states Kunst et al. (2018), and (ii) a construction combining information about both the open and closed systems that leads to modified topological invariants akin to those in the Hermitian realm Yao and Wang (2018); Yao et al. (2018). While seemingly distinct we elucidate the equivalence of these two approaches, which from a different angle provide accurate predictions for generic NH systems. We also give a brief overview of complementary works relating to NH BBC Herviou et al. (2019a); Lieu (2018a); Brzezicki and Hyart (2019); Esaki et al. (2011); Lee and Thomale (2019); Borgnia et al. (2019); Zirnstein et al. (2019); Imura and Takane (2019); Edvardsson et al. (2019a); Kunst and Dwivedi (2019); Yao et al. (2018); Yokomizo and Murakami (2019); Yang et al. (2019c); Song et al. (2019b).
iii.2.1 Biorthogonal bulk-boundary correspondence
Biorthogonal quantum mechanics. To discuss the biorthogonal BBC introduced by Kunst et al., 2018 in a self-contained manner, we very briefly recall basic elements of biorthogonal quantum mechanics (QM) (see Brody, 2013 for a pedagogical review). Biorthogonal QM can be seen as a generalization of ordinary QM by allowing for the treatment of NH observables, and reduces to ordinary QM upon restoring Hermiticity. As mentioned in Section I, an NH Hamiltonian in general has inequivalent right and left eigenvectors, and , respectively, such that its eigenvalue equations read
where the latter expression is alternatively written as . As is clear from our minimal example in Section I, the left and right eigenvectors generally do not form an orthonormal set with the standard inner product [cf. Eq. (3)]. However, the essence of biorthogonal QM is that, away from exceptional degeneracies, the sets and form a useful biorthogonal basis by demanding
(23) |
As we shall see below this seemingly innocent change in normalization condition has profound implications since the left and right eigenstates can be strikingly different and may even localize at opposite boundaries of the system. An immediate and important consequence is that the energy eigenvalues of an NH Hamiltonian are given by its expectation value with respect to the right and left wave functions, i.e.,
(24) |
Expectation values of the form of Eq. (24) are known as biorthogonal expectation values, and play a central role in understanding the dynamics of NH models.
Biorthogonal BBC. In the following, we discuss how the biorthogonal formalism can be used to construct a variant of the BBC that remains intact for NH systems with a line gap, and reduces to cBBC in the Hermitian limit. This approach, coined biorthogonal BBC, was introduced by Kunst et al., 2018, showing that one way to qualitatively and quantitatively understand the physics of NH models with OBC is by making use of biorthogonal QM.
To illustrate this method, we make explicit use of the example in Eq. (21). By Kunst et al., 2018, it was shown for the Hamiltonian in Eq. (21) with OBC that it is possible to write the following ansatz for the zero-energy state, which is exponentially localized and has nonzero weight on the sublattices only,
(25) |
where () is the normalization factor of the right (left) wave function, labels the unit cell with a total of unit cells, and creates a state in the vacuum on sublattice in unit cell . Remarkably, the localization factors and are different
(26) |
and hence, depending on the parameter values, the left and right states can be localized on either the same or at opposite boundaries. It is worth noting that the possibility of having the left and right states localized at opposite boundaries imply that the biorthogonal normalization condition [Eq. (23)] becomes radically different from the standard normalization condition familiar from the Hermitian realm.
To study the localization of the zero-energy states in the lattice, the biorthogonal expectation value of the projection operator with onto each unit cell is computed, and leads to for the wave functions in Eq. (25). According to this expression, the zero-energy state is thus a bulk state when , i.e., when it is equally localized to all unit cells, while it is exponentially localized to when , and disappears into the bulk for . This indeed corresponds to what we see in the band spectrum in the inset of Fig. 6(b) up to finite-size corrections: The bulk gap closes when , while the in-gap zero-energy states exist for . Note that identical results are found when considering the biorthogonal expectation value of the projection operator with respect to the zero-energy state localized on the sublattices at the end . Thus, determines whether boundary states exist, defining the notion of a biorthogonal BBC. By contrast, the ordinary expectation values (based on left and right eigenstates, respectively) yield , and , respectively, for the wave function in Eq. (25). Both these expectation values coincidentally predict gap closings in the PBC spectrum, and thus fail to correctly predict the formation of zero-energy edge modes when cBBC is broken.
Biorthogonal polarization. Generalizing the insights gained from the aforementioned quantity , Kunst et al., 2018 introduced the biorthogonal polarization
(27) |
From this expression, it is straightforward to see that equals one in the presence of end states, i.e., when in the above discussion, and zero when no such states exist, i.e., when . jumps when the gap closes corresponding to . As such, the value of the biorthogonal polarization accurately predicts the presence of boundary states inside the bulk gap, and can thus be interpreted as a real-space invariant.
We note that the biorthogonal polarization is equal for models that are related to each other via unitary transformations acting locally, e.g., for the nonreciprocal SSH model equals for Lee’s model discussed in Sect. III.1.1 Edvardsson et al. (2019b).
Generalizations. As is pointed out by Kunst et al., 2018, the wave function solution in Eq. (25) can be straightforwardly generalized to a large family of lattice models with any dimension such as NH Chern insulators in two dimensions. Further generalizations to higher-order boundary states of NH models work analogously Edvardsson et al. (2019a): In each case determines the existence of boundary states and accurately predicts the occurrence of phase transitions. It has also been verified that the definition of the biorthogonal polarization can be naturally extended to models with multiple boundary states on one boundary Edvardsson et al. (2019b).
We emphasize that the biorthogonal polarization defined in Eq. (27) is not limited to solutions of the form given in Eq. (25), but can be computed for any boundary state in generic NH models that do not afford exact analytical solution Kunst et al. (2018). This makes the biorthogonal BBC a general principle for NH topological models, which recovers the cBBC where applicable.
Lastly, we note that while right wave functions are most naturally accessible in experiment, Schomerus, 2020 proposed in a recent theoretical work that it is also possible to probe left wave functions as well as the biorthogonal contribution of both right and left wave functions beyond the spectral properties when measuring the response functions to external perturbations in robotic metamaterials such as the ones studied in Brandenbourger et al., 2019; Ghatak et al., 2019 (see Sect. IV.1.2 for a more detailed discussion).
iii.2.2 Non-Bloch bulk-boundary correspondence
An alternative strategy for finding a generalized BBC is presented in Yao and Wang, 2018; Yao et al., 2018 and further expanded by Yokomizo and Murakami, 2019 and Yang et al., 2019c. There, a generalized BZ is constructed to include information, which in the case of cBBC is not contained in the standard Bloch bands, pertinent for the accurate definition of bulk topological invariants. The key idea of this approach is that a state in unit cell of a model with OBC, , can be written as , where . Solutions for in terms of the hopping parameters and energy eigenvalues are then found by solving the eigenequations using this ansatz Yao and Wang (2018). From these solutions, it is possible to derive the generalized BZ , and to find expressions for the boundary states, as we briefly review in the following.
The generalized BZ is found by looking at the condition for obtaining the bulk bands Yokomizo and Murakami (2019). Ordering the solutions according to , where with degrees of freedom and the range of hopping, the bulk states are retrieved by demanding
(28) |
This condition is derived by assuming that the system size is large, and that the energy states are densely distributed. The complex-valued trajectories of and then form the generalized BZ , which in the case of Hermitian or cBBC-preserving NH Hamiltonians simply reduces to the unit circle, i.e., to the conventional one-dimensional BZ.
When , the bulk states exhibit the NH skin effect: They localize to the left boundary for and to the right boundary for . As already mentioned, it is also possible to find models with skin states that are localized to opposite boundaries (cf. Sect. III.1.2), in which case part of the generalized BZ lies inside the unit circle, and part of it outside Song et al. (2019b).
If the energy of possible topological boundary modes is known, it is possible to find a solution for these states by plugging into the solutions Yao and Wang (2018). The bulk-band gap then has to close when , i.e., when the topological boundary state merges with the bulk bands. We note that this merging condition () is equivalent to the condition found within the biorthogonal framework Kunst et al. (2018): Indeed, the anistropic SSH model in Eq. (21) is also studied by Yao and Wang, 2018 leading to equivalent results for the topological boundary states as well as their attachment to the bulk bands.
As the energy of the boundary states is not usually known, however, an alternative way to find the band-gap closing is to make use of what by Yao and Wang, 2018 is called non-Bloch topological invariants: Replacing with , or equivalently, applying a shift in the wave vector , in the Bloch Hamiltonian leads to the so-called “non-Bloch Hamiltonian” defined on the generalized BZ, and allows for the computation of non-Bloch topological invariants, which correctly predict the existence of topological boundary states. Indeed, it was exemplified that a winding number derived for on the generalized BZ correctly predicts the existence of the zero-energy end states for the model in Eq. (21) by Yao and Wang, 2018, and a variation of the model by Yokomizo and Murakami, 2019. Furthermore, Yang et al., 2019c elaborated on the geometrical interpretation of the generalized BZ, Lee et al., 2019b derived physical responses based on this picture and Yao et al., 2018 introduced a non-Bloch Chern number which accurately predicts the existence of chiral edge states.
We note that the ansatz for the wave function, which is the basis of the generalized BZ construction, can be seen as a generalization of the usual ansatz in Hermitian systems, obtained by shifting the wave vector according to . Indeed, for Hermitian and cBBC-preserving systems, , such that and Bloch’s theorem is retrieved. In this case, the condition in Eq. (28) is trivially satisfied for all , showing that also this approach connects to the well-established Hermitian limit in the expected way.
iii.2.3 Complementary approaches
We now give a brief overview of complementary perspectives and approaches to BBC in NH systems reported in recent literature.
Refinements. The NH BBC developed by Kunst et al., 2018; Yao and Wang, 2018; Xiong, 2018 has been refined and corroborated by a number of recent studies. As discussed in Section III.2.1 for the biorthogonal approach it is beneficial to have access to exact solutions to understand the properties of NH models. As a complementary approach to obtain such exact solutions transfer-matrix methods have been introduced in the context of NH models by Kunst and Dwivedi, 2019. There, one of the central results is that the determinant of the transfer matrix associated with a given NH hopping model plays a crucial role in determining whether cBBC is broken: Namely, when the transfer matrix is unimodular, i.e., , the PBC and OBC spectra are equivalent, and bulk states in the OBC case behave in the ordinary fashion. When , on the other hand, more interesting properties arise: The bulk spectra for PBC and OBC are different, while the norm of the bulk states in the OBC case is proportional to , with labelling the supercell, thus clearly signaling the NH skin effect. It is possible to tune between the bulk spectra by applying a shift to the crystal momentum, i.e., when , where this shift in the Bloch momentum is equivalent to the one found by Yao and Wang, 2018 thus corroborating the generalized BZ approach. The transfer-matrix method also corroborates the findings from the biorthogonal approach: The eigenvalues of the transfer matrix for the boundary states, which correspond to the decay coefficients of said boundary states, naturally lead to the definition of a merging condition equivalent to the one found by Kunst et al., 2018 (). Additionally, transfer matrices can also be used to determine the appearance of EPs in the OBC spectrum. Indeed, when it is possible to hop only in one direction and EPs with an order scaling with system size naturally show up in the OBC spectrum (see also the discussion in Sect. III.1.2).
The biorthogonal and non-Bloch frameworks are further expanded by Lee and Thomale, 2019, where a complex flux effectively interpolating between PBC and OBC is used Hatano and Nelson (1996), which is equivalent to tuning the value of the complex part () of the complex momentum () as introduced by Yao and Wang, 2018. The insertion of the complex flux allows for the derivation of a condition for the existence of bulk, and more particularly, skin states akin to the one in Eq. (28).
The BBC of NH models is also studied by making use of Green’s functions, or more specifically, boundary Green’s functions in Borgnia et al., 2019; Zirnstein et al., 2019 to find topological phase diagrams. In particular, Zirnstein et al., 2019 used this machinery to study NH Dirac fermions in one dimension, and a nonzero winding number [cf. Eq. (6) with replaced by ] is found to lead to a spatial growth of the bulk Green’s function signaling a breakdown of cBBC and the occurrence of the NH skin effect. This relation between a nontrivial winding number and the appearance of skin states has indeed been elaborated upon at the level of dynamic matrices Wanjura et al. (2019) as well as at the Hamiltonian level in recent works Zhang et al. (2019a); Okuma et al. (2019) (cf. Sect. III.1.2). Borgnia et al., 2019 found edge modes by computing the in-gap zeros of the doubled boundary Green’s function, where the input Hamiltonian is of the form of Eq. (19). There, by studying the Green’s function in this framework, a classification of NH models in terms of their gaps is found, thus extending the results from Zirnstein et al., 2019.
As a complementary approach, Imura and Takane, 2019 proposed modified periodic boundary conditions (mPBC) to restore BBC in NH systems. The key idea is that the mPBC incorporate the NH skin states directly into a modified periodic model from which it is then possible to compute topological invariants that accurately predict the existence of boundary states in the case of OBC. The mPBC by Imura and Takane, 2019 bear some similarity to the argument of imaginary flux threading by Lee and Thomale, 2019: The mPBC are implemented through the inclusion of prefactors and in the Hamiltonian that connects the two ends and , while the flux threading essentially introduces a similar prefactor to the Hamiltonian. While this mPBC method seems very similar to the non-Bloch BBC introduced by Yao and Wang, 2018, there is nevertheless a subtle difference: To establish the non-Bloch BBC reference is made to a system with OBC to find the relevant needed to compute the non-Bloch topological invariants. In the context of mPBC, by contrast, no reference to OBC is required to find the topological invariants.
Alternative perspective: singular value spectrum. Herviou et al., 2019a, b; Porras and Fernández-Lorenzo, 2019 proposed to infer the topological phase diagram and the existence of boundary modes by a singular value decomposition (SVD). There, the role of the eigenvalues of an NH matrix is replaced by its singular values that do not exhibit the aforementioned spectral instability, and the counterpart of the eigenvectors may be directly inferred from the transformation matrices of the SVD. This not only allows for the stable computation of topological invariants, which are constructed by making use of a generalized flattened singular decomposition, but also for a generalization of the concept of the entanglement spectrum to realm of NH models Herviou et al. (2019a). However, the SVD approach leads to a restoration of cBBC, even in models where cBBC is found to be broken when studying the eigenvalue spectrum. Thus, the exotic features displayed by cBBC-breaking models are not fully captured within the SVD perspective.
Symmetries. The influence of symmetries on BBC in NH has been widely studied Esaki et al. (2011); Lieu (2018a); Brzezicki and Hyart (2019); Kunst and Dwivedi (2019); Kawabata et al. (2019c) (see also Sect. III.1.2), and cBBC has been shown to be preserved in a number of symmetric NH models. For example, Esaki et al., 2011 showed that even though the spectrum of NH systems generically is complex, it is possible to find topological invariants from the Bloch Hamiltonian that accurately predict the existence of boundary states in the real part of the spectrum for specific lattice models, which either feature pseudo-Hermiticity [cf. Eq. (17) with ] or time-reversal symmetric with the time-reversal operator [cf. Eq. (14)]. Remarkably, in some NH systems even the TRS of type leads to a generalized Kramers’ theorem Sato et al. (2012). A related form of the pseudo-Hermitian symmetry has been investigated by Brzezicki and Hyart, 2019, where a special form of NH chirality is studied, i.e., . By considering one-dimensional models in the presence of this symmetry, a hidden Chern number can be defined, which determines the number of end states whose real part of the energy is zero. There, the imaginary part of the energy is used as a second dimension, which offers a new perspective on the definition of topological invariants in NH models. Leykam et al., 2017; Lee, 2016 also studied chiral symmetry and find half-integer winding numbers characterizing the EPs in the spectum. Lieu, 2018a studied both chiral symmetry and PT symmetry in the context of NH variations to the SSH model, and topological invariants derived from the Bloch Hamiltonian have been found for both cases. More specifically, a global invariant can be defined in the -symmetric case, while a quantized complex Berry phase exists in the case with chiral symmetry. With the more recent studies of the NH model in Eq. (21), which is also chirally symmetric, however, we know that such a complex Berry phase cannot always be found, or at least needs to be modified, for example by using the techniques developed in the non-Bloch setting Yao and Wang (2018); Yao et al. (2018); Yokomizo and Murakami (2019).
iii.2.4 Summary: A unified picture
Having reviewed various complementary approaches to re-establishing the BBC in NH systems, we now briefly summarize the different methods by drawing a unified picture. Whereas the main approaches introduced by Kunst et al., 2018 and by Yao and Wang, 2018; Yao et al., 2018, respectively, have very different vantage points, we stress that they lead to identical predictions in full agreement with a wide range of explicit model calculations.
While Kunst et al., 2018 took direct cue from the properties of systems with OBC and examine the (de-)localization transitions of the biorthogonal wavefunctions, Yao and Wang, 2018; Yao et al., 2018 instead augmented the Bloch Hamiltonian with information from the OBC leading to a generalized Brillouin-zone (often called non-Bloch) description that relates more directly to the familiar picture of the cBBC in terms of topological invariants. The biorthogonal approach, on the other hand, offers additional physical insights in terms of a quantized polarization and reveals the key role played by the interplay between left and right wave functions, a distinction that is inherently NH. Taken together they thus offer a comprehensive framework and physical intuition for cBBC-breaking NH models. Moreover, despite their differences in appearance, these approaches do share the emphasis on a wave-function ansatz, which are also utilized and expanded on in Kunst and Dwivedi, 2019; Lee and Thomale, 2019; Imura and Takane, 2019.
Several recent works have corroborated and elucidated the non-Bloch approach either by making the Bloch momentum complex Yokomizo and Murakami (2019); Kunst and Dwivedi (2019), or by applying mPBC Imura and Takane (2019). Complementing this perspective Borgnia et al., 2019; Zirnstein et al., 2019 made a direct connection to OBC thus being conceptually more in line with biorthogonal approach while work by Kunst and Dwivedi, 2019; Lee and Thomale, 2019 interpolates between PBC and OBC cases, and may as such be seen as a bridge between the approaches using Bloch Hamiltonians and those using OBC descriptions.
Iv Physical platforms
We now give an overview of experimental platforms for the observation of NH topology, reflecting the wide range of physical implications of the NH phenomena discussed above.
iv.1 Non-Hermitian wave equations: From classical mechanics to quantum walks
Intense research efforts in recent years have unraveled classical analogues of topological phases in a rich variety of settings ranging from photonics Haldane and Raghu (2008); Raghu and Haldane (2008); Ozawa et al. (2019) to electric circuits Ningyuan et al. (2015); Albert et al. (2015); Lee et al. (2018a) and mechanical systems Huber (2016); Kane and Lubensky (2013). Guided by the intuition of closed non-dissipative systems such analogies were initially established for (nearly) Hermitian systems. However, in all of these settings non-Hermiticity is actually ubiquitous reflecting the fundamental role of dissipation. Indeed, the profound conceptual advances in the understanding of NH topological phenomena discussed in this review have been closely accompanied by corresponding experiments in all of the aforementioned platforms. In these classical platforms the analogy with Hamiltonian QM is manifested in a number of different ways: Some settings directly mimic the time-dependent Schrödinger equation as in coupled optical wave guides, while in, e.g., photonic crystals and acoustic systems the eigenmode problem is tantamount of the Bloch problem in quantum mechanics. Similarly, in, e.g., robotic (mechanical) metamaterials the analogy to Hamiltonian QM is directly to an asymmetric dynamical matrix, while in electrical circuits the analogy is on the level of response functions. While we refer to the original work for full details we here outline some of the key ideas.
iv.1.1 Photonics
Photonics is arguably the area in which NH topology has so far found most applications. For a recent in-depth account on (mostly Hermitian) topological photonics we refer the reader to the recent review Ozawa et al. (2019). We here highlight a few systems with particular relevance to the genuinely NH phenomena.
Let us begin with photonic crystals in which the basic idea is to create metamaterials with spatially varying but periodic dielectric permittivity and magnetic permeability Joannopoulos et al. (2011). In this setting the electrodynamic eigenmodes of Maxwells equations are subject to Bloch’s theorem in a very similar way as it applies to electrons in crystalline solids. Inspired by the seminal theoretical proposal for photonic analogues of quantum Hall states due to Haldane and Raghu, 2008; Raghu and Haldane, 2008, and subsequent refinements by Wang et al., 2008, classical analogues of topological states have been realized in gyromagnetic photonic crystals, which explicitly break time-reversal symmetry Wang et al. (2008); Lu et al. (2013). In these systems gain and loss is ubiquitous and NH topological phenomena have been experimentally realized including a spectacular observation of Fermi arcs connecting EPs Zhou et al. (2018) as theoretically described in Sec. II.2, as well as a demonstration of one-sided invisibility in -symmetric metamaterials Feng et al. (2013) predicted to occur in -symmetric materials operating at an EP Kulishov et al. (2005); Longhi (2011); Lin et al. (2011); Jones (2012), which had also been shown in a scattering experiment Regensburger et al. (2012) (cf. Sect. IV.2). Recent theoretical work has suggested that the Maxwell waves existing on the interfaces separating lossless media with different signs in the permittivity and permeability have topological properties, which are related to the properties of an NH helicity operator Bliokh et al. (2019) thus further highlighting the NH character of photonic crystals.
Photonic crystals belong to the larger experimental platform of optical microresonators, also known as microcavities Vahala (2003). The performance of such resonators is captured by the factor, which is proportional to the lifetime of a photon inside the cavity, and is strongly dependent on the properties of the interface between the cavity volume and the outside. Interestingly, a coupled-microresonators setup with auxiliary resonators with gain and loss has been proposed to realize the Hatano-Nelson model Longhi et al. (2015), whereas active steering of topological light has been demonstrated in two-dimensional lattices of microresonators with reconfigurable gain and loss domains Zhao et al. (2019). One prominent example of optical microcavities with very high factors is that of whispering-gallery-mode resonators (WGMR) Lefèvre-Seguin and Haroche (1997); Gorodetsky et al. (1996); Vernooy et al. (1998b); Knight et al. (1995); Vernooy et al. (1998a), which derive their name from their acoustical counterpart: Electromagnetic waves are captured in the cavity because of total internal reflection.
Recently, NH experimental setups of such WGMRs were shown to exhibit unidirectional lasing Peng et al. (2016, 2014), single-mode lasing in -symmetric setups Feng et al. (2014); Hodaei et al. (2014), and enhanced sensitivity against perturbations in cavities operating at second-order EPs Chen et al. (2017) due to the nonanalytic behavior of their dispersion Wiersig (2014). Similar behavior has also been demonstrated in higher-order EPs realized in an arrangement of coupled micro-ring resonators Hodaei et al. (2017).
Optical resonators operating at microwave frequencies are known as microwave cavities, and recently, the dynamical encircling of second-order EPs has been studied in such a setup revealing clear experimental signatures of mode switching Doppler et al. (2016) (as we already saw in the minimal example in Sect. I). Additionally, open microwave disks form the perfect platform to study the quantum-classical correspondence in open systems, and experiments on such models demonstrate that classical quantities can describe their quantum properties and vice versa Potzuweit et al. (2012); Lu et al. (1999); Pance et al. (2000); Barkhofen et al. (2013).
Coupled wave guides provide another versatile platform which, instead of simulating static properties, directly emulates the time-evolution of tailor-made lattice models Davis et al. (1996); Longhi (2009); Christodoulides et al. (2003). The waveguides are routinely inscribed in silica glass using femtosecond lasers and have the additional appealing feature that they operate well at optical frequencies visible to the human eye Szameit and Nolte (2010). Here, Maxwell’s equations describing the propagation of light in the direction amount to the paraxial equation
which is formally identical to the two-dimensional Schrödinger equation with the propagation direction playing the role of time , and the “wave function”, , is the envelope of the electric field polarized along such that is assumed to be slowly varying in the sense that with . Crucially, the effective potential can be tailor-made by carving waveguides using accurate femtosecond lasers, which create a strong spatial dependence of the local refractive index . In the limit of spatially sharp carving and weak evanescent coupling between the waveguides this system is accurately modelled by a tight-binding Hamiltonian whose hopping parameters depend on the setup and on the wavelength, , of the light. This setup has been harnessed to emulate a large number of Hermitian topological phases Rechtsman et al. (2013); Noh et al. (2015, 2018); El Hassan et al. (2019), and including staggered patterns of gain and loss in the wires, the time-evolution of effectively NH models has also been successfully simulated. Notably this includes the experimental realization of exceptional rings Cerjan et al. (2019) (cf. Sec. II.2), defect states in NH SSH chains Weimann et al. (2017) (cf. Sec. III.1.4), topological phase transitions Zeuner et al. (2015), and -symmetric flat bands Biesenthal et al. (2019), whereas a study of the stability of corner states against gain and loss has also been proposed Özdemir and El-Ganainy (2019). Here it is worth noting that passive systems with only staggered loss, e.g., from wave guides of alternating quality, is sufficient to generate such phases—although the energies are confined to the lower complex half-plane a global shift can make the system effectively -symmetric in a description, where the less lossy waveguides thus effectively experience gain Guo et al. (2009); Ornigotti and Szameit (2014); Weimann et al. (2017); Feng et al. (2013); Kremer et al. (2019). In fact, a truly -symmetric system is realized by making use of optical fibres by Regensburger et al., 2012, where the use of optical amplifiers and modulators allows for the realization of a -symmetric structure in the temporal domain.
iv.1.2 Mechanical systems
Mechanical systems represent another experimental medium with which NH phases can be realized. One type of such systems is provided by mechanical metamaterials (see Huber, 2016; Bertoldi et al., 2017 for recent reviews), which can be described as networks consisting of masses that are connected via springs of rigid beams, and are governed by Newton’s equations. Drawing from a connection between Newton’s second law and the Schrödinger equation—the equations of motion for a system of coupled oscillators with the oscillators, describing the non-dissipative coupling between position and velocity, and the dynamical matrix capturing the forces between oscillators, can be recasted into the following Hermitian eigenvalue problem
as detailed in Kane and Lubensky, 2013; Süsstrunk and Huber, 2015; Huber, 2016—, it is possible to realize topological phases featuring phononic boundary states in these setups. Indeed, topological phononic modes, which are classified by Süsstrunk and Huber, 2016, have been reported to appear at the boundaries of isostatic lattices build with springs Kane and Lubensky (2013), at the boundaries in models consisting of rotors and rigid beams Chen et al. (2014), at dislocations in kagome lattices consisting of rigid plates Paulose et al. (2015), and as helical boundary states in a setup consisting of pendula Süsstrunk and Huber (2015). When the masses are replaced by gyroscopes, one obtains a so-called gyroscopic metamaterial, which has been shown to host acoustic boundary waves analogues to the edge states of the quantum Hall effect Nash et al. (2015); Wang et al. (2015).
Inspired by these results and the connection between the dynamical matrix and the Hamiltonian description in these setups, one can conceive NH phononic phases: Starting from a generic NH Hamiltonian matrix with offdiagonal elements and , the dynamical matrix is defined as Ghatak et al. (2019). This way of writing the dynamical matrix is in close analogy with the method presented by Kane and Lubensky, 2013, who study isostatic lattices, which are mechanically critical in the sense that they are near collapsing: The dynamical matrix associated with the lattice is written as , such that by taking the ‘square root’ one obtains the associated Hamiltonian matrix, which has and as its offdiagonal elements. Such a dynamical matrix for NH Hamiltonians (), which is asymmetric (), has been experimentally realized in robotic metamaterials Brandenbourger et al. (2019), which combine robotics and active materials through building lattices consisting of mechanical rotors, control systems and springs. In such setups, the NH skin effect has been observed in a nonreciprocal realization Brandenbourger et al. (2019) as well as in a model similar to the anisotropic SSH chain in Sect. III.1.1 Ghatak et al. (2019). Both these experiments thus probe the right wave functions of the model that they investigate. In a recent work, Schomerus, 2020 showed by making use of response theory that it is possible to also probe the left wave functions in these setups: Whereas right wave functions specify the spatial distribution of the response of the setup to an external excitation, the information on the strength of this response with respect to where the perturbation is located is captured by the left wave functions. When considering the overall response, which includes contributions from both the right and left wave functions, Schomerus, 2020 showed that the NH skin effect of the zero mode is related to a phase transition at which the sensitivity to perturbations becomes critical in the sense that it diverges. The inherent biorthogonality of these systems thus leaves clear experimental signatures beyond the characteristic energy spectra. These two experiments, Brandenbourger et al., 2019; Ghatak et al., 2019, also prompted the study of the NH skin effect in elastic lattices with non-local feedback interactions: Rosa and Ruzzene, 2020 found that non-local control allows for bulk waves to localize at different boundaries, such that a judicious choice of interactions can result in corner localization as is illustrated in two-dimensional models. Scheibner et al., 2019 showed that an anti-symmetric dynamical matrix, , can be realized in mechanical metamaterials with odd elasticity, which occurs due to non-energy-conserving microscopic interactions in active media. The odd elasticity is predicted to facilitate the onset of exceptional points for an overdamped lattice as well as to sustain an elastic engine cycle for an overdamped wave Scheibner et al. (2019), to allow for the appearance of bulk elastic waves at the boundaries of one- and two-dimensional metamaterials Zhou and Zhang (2019), and to host a topological phase transition mediated by the annihilation of exceptional rings in active as well as gyroscopic metamaterials with gain and loss Scheibner et al. (2020). Another recent realization of an NH phase in mechanical metamaterials is presented by Yoshida and Hatsugai, 2019, where exceptional rings are proposed to appear in mechanical metamaterials with friction.
Another type of mechanical system, which can be used to realize topological phases, is that of phononic or acoustic metamaterials first proposed by Kushwaha et al., 1993. Such materials consist of elastic composites that are arranged in a periodic fashion, and have been shown to host phononic edge states in microtubules Prodan and Prodan (2009), quantum-spin-Hall edge states in the form of elastic waves Mousavi et al. (2015), and surface acoustic waves with negative refraction index on the surfaces of a phononic version of a Weyl semimetal He et al. (2018). Acoustic waves may also propagate through fluids, and a setup consisting of rotating fluids arranged in a crystal was predicted to realize the chiral edge states of the quantum Hall effect Yang et al. (2015). This experimental platform can be used to realize NH phases through the judicious implementation of gain and loss. Indeed, Shi et al., 2016 realized a -symmetric model, where gain is implemented via coherent acoustic sources, in which they gain full control of the EP and the accompanying unidirectional transparancy. A -symmetric acoustic metamaterial was also realized by Aurégan and Pagneux, 2017 in an airflow duct with gain and loss implemented through the scattering of acoustic waves of diaphragms. Similarly, Rivet et al., 2018 showed that acoustic waves with constant pressure can exist in acoustic waveguides with gain and loss, while Zhu et al., 2018 realized an EP in a lossy acoustic system and demonstrated unidirectional propagation. Additional theoretical proposals exist for the realization of -symmetric second-order topological phases in acoustic metamaterials with gain and loss Rosendo López et al. (2019); Zhang et al. (2019b), and invisible acoustic sensors with PT symmetry Fleury et al. (2015).
iv.1.3 Electric circuits
Electric circuits provide another classical platform for the realization of NH topology Ningyuan et al. (2015); Albert et al. (2015). Here, instead of being a property of the Hamiltonian, one directly studies response functions, where capacitors and inductors act as Hermitian elements, and resistors as well as amplifiers are anti-Hermitian. As a specific example, a current depending on frequency flowing through a node is governed by the relation
where and are the input current and potential at node , respectively, and is the admittance matrix, or equivalently, the inverse impedance matrix , where with is the admittance between nodes and , and is the admittance between node and the ground Ningyuan et al. (2015). This relation can be derived by making use of current conservation, i.e., the total input current needs to equal the total output current, and amounts to Kirchhoff’s circuit laws.
The periodicity of the electric circuit structures allows for the use of Bloch’s theorem to find wave functions, while the band structure of the circuits corresponds to the eigenvalues of the admittance up to a prefactor. As such, one can interpret the admittance matrix as a Hamiltonian matrix. Through the clever arrangement of capacitors, inductors, and other electronic tools available in this toolbox, it is thus possible to design circuits, which mimic the physics of topologically nontrivial models. This idea was first introduced by Ningyuan et al., 2015, and has been used to build topological circuits whose band structures, i.e., admittance eigenvalues, realize the band topology of the Hofstadter model Ningyuan et al. (2015); Albert et al. (2015), also in the Möbius strip configuration Ningyuan et al. (2015). More recently, the SSH chain and a two-dimensional extension thereof as well as a Weyl semimetal spectrum have been reported by Lee et al., 2018a, whereas corner states were realized in two-dimensional setups in Imhof et al., 2018.
These realizations of Hermitian topological phases in electric circuits have paved the way to the fabrication of NH versions thereof. Indeed, by making use of resistors and amplifiers, the NH SSH model in Eq. (21) was realized very recently by Helbig et al., 2019 corroborating the theoretical predictions. The NH skin effect was subsequently also measured by Hofmann et al., 2019b. Additional proposals exist for the realization of NH Chern insulators Hofmann et al. (2019a); Ezawa (2019b), higher-order topological models with NH skin states localized to lower-dimensional boundaries Ezawa (2019d, c), a quantum walk simulation Ezawa (2019a) (see Sect. IV.1.4), the realization of three-dimensional Seifert surfaces in four-dimensional circuit setups Li et al. (2019a), as well as the implementation of a pseudo-magnetic field to probe exceptional Landau levels in NH Dirac and Weyl systems Zhang and Franz (2019).
iv.1.4 Quantum walks
Quantum walks, which are rather an experimental concept than being limited to a specific platform, provide another means to simulate and probe NH topological phases. Quantum walks can be seen as the quantum version of classical random walks, where the “coin flip”, which introduces the classical randomness through determining the trajectory of a particle, is replaced by a coin operator acting on the internal degrees of freedom of a particle, also known as the “walker”. The concept of a quantum walk was first introduced by Aharonov et al., 1993, and quantum walks have been realized in several experimental platforms such as trapped atoms Karski et al. (2009), trapped ions Zähringer et al. (2010); Schmitz et al. (2009), optical fiber networks Schreiber et al. (2010); Broome et al. (2010), and nuclear-magnetic resonances Ryan et al. (2005).
The dynamics of a quantum walk is captured by a Floquet operator , which depends on the coin operator and is related to a time-independent effective Hamiltonian via . Through a suitable choice of , the effective Hamiltonian can be made to be topologically nontrivial resulting in the appearance of topological phases in quantum walks as predicted in theory Kitagawa et al. (2010); Asbóth (2012) and shown experimentally in discrete-time quantum walks Kitagawa et al. (2012); Cardano et al. (2016); Barkhofen et al. (2017); Ramasesh et al. (2017); Flurin et al. (2017) (see Wu et al., 2019 for a recent review), where the Floquet operator U is applied to the walker at discrete time steps.
By instead considering a non-unitary Floquet operator , the effective Hamiltonian of the model is NH, and it is thus possible to study NH phases. This idea was first introduced by Rudner and Levitov, 2009 for an NH SSH model with loss on every other site thus realizing a passive version of a -symmetric SSH chain, where it is shown that the average displacement of the particle is quantized and associated with a topological invariant. Experiments on such non-unitary quantum walks reveal the existence of topological edge states at domain walls in a -symmetric SSH chain in an optical setup with balanced gain and loss Xiao et al. (2017), as predicted in theory Mochizuki et al. (2016). Zhan et al., 2017 detected topological invariants, Wang et al., 2019b studied dynamic quantum phase transitions in -symmetric system, Wang et al., 2019a observed skyrmions in a -symmetric non-unitary quantum walk, and Longhi, 2019b predicted the appearance of the NH skin effect and a symmetry-breaking phase transition in a PT-symmetric discrete-time non-unitary quantum walk. Models with anisotropic hoppings have also been realized in a discrete-time non-unitary quantum-walk setup, where the NH skin effect has been explicitly detected Xiao et al. (2019).
iv.2 Quantum many-body systems
While most early applications of NH topology were based on classical physics and single-particle quantum mechanics, non-Hermiticity also plays an important role in genuinely quantum mechanical many-particle systems. Indeed, the study of NH Hamiltonians in this context has a long history with applications, e.g., in nuclear and atomic physics Rotter (2009); Majorana (1931a, b); Fano (1961); Breit and Wigner (1936); Feshbach (1958); Feshbach et al. (1954). More recently, the relevance of these Hamiltonians to topological phases has been investigated in several quantum many-body platforms as outlined below.
iv.2.1 Open systems
Quantum master equations. The most natural source of non-Hermiticity in quantum many-body systems is quantum dissipation as induced by coupling the system to its environment. A realm of direct relevance are quantum optical setups and ultracold atomic gases, where experiments are often carried out in the regime of a weak coupling to a Markovian reservoir represented by the continuum of surrounding electromagnetic field modes. In such situations, the relevant equation of motion for the reduced density matrix of the open system is the Lindblad master equation Lindblad (1976)
(29) |
where the jump operators account for the coupling to the environment. Mostly focusing on the case of pure dissipation (), the dissipative preparation of topological states within the full Lindblad setting has been investigated Diehl et al. (2011); Bardyn et al. (2013); Budich et al. (2015); Tonielli et al. (2019); Goldstein (2019). However, due to the complexity of the Lindblad master equation, a different approach is desirable for obtaining an intuitive understanding of the interplay between coherent quantum dynamics, dissipation, and topology in complex quantum many-body systems. To this end, one useful approach is to note that the Lindblad equation can conveniently be written as where the effective NH Hamiltonian
(30) |
describes the dynamics at short times Carmichael (2014). At longer times the so-called jump (or recycling) term, , accounting for the actual occurrence of quantum jumps can typically no longer be neglected. In the general situation this thus leads to decoherence (hence mixed states) while the effective non-Hermitian description is by construction in terms of (less general) pure states. Nevertheless, the relevance of NH Hamiltonians for Lindblad systems reaches far beyond the obvious realm at short lifetimes: It is easy to construct intriguing examples where the steady state of the Lindblad equation is identical to the state resulting from the non-unitary time evolution of an effective NH Hamiltonian. An especially simple and constructive way of achieving this is to reverse engineer models using the condition Diehl et al. (2011), which in effect can target, e.g., the ground state of a model Hamiltonian by a suitable choice of the Lindblad jump operators. This approach is particularly well suited for preparing topological phases that quite generically have parent Hamiltonians composed of non-commuting terms that can nevertheless be simultaneously minimized. This may serve as an efficient way of harnessing dissipation and the intuition from NH Hamiltonians to realize essentially Hermitian topological phases. It is also worth noting that the effective Hamiltonian (30) has eigenvalues in the lower complex half-plane . This highlights the fact that the Lindblad equation, even in the regime accurately captured by Eq. (30), imposes a fundamental constraint on eligible NH Hamiltonians as compared to the fully generic case (cf. Section II.4).
For Gaussian systems described by a Lindblad equation that is quadratic in the field operators, there is another way of systematically deriving an effective NH description in terms of a damping matrix Prosen (2010); Eisert and Prosen (2010). Complementary to the above , the NH matrix governs how deviations from the steady state are damped out, thus describing the long-time limit of the Lindblad equation. Interestingly, these two effective NH matrices have been shown to generally differ in their topological properties Song et al. (2019a). In the context of Gaussian Lindbladians, intriguing genuinely NH phenomena have recently been discovered Song et al. (2019a); Lieu et al. (2019); Hatano (2019). A salient example along these lines is that the remarkable phenomenology of the non-Hermitian skin effect carries over, mutatis mutandis, to the more fundamental Lindblad setting Song et al. (2019a) where it had previously been overlooked. Moreover, exceptional points also appear naturally within the Lindblad master equation framework Hatano (2019), and certain classes of quadratic Lindblad operators admit a classification analogous to that of NH Hamiltonians Lieu et al. (2019).
Material junctions in quantum transport setups provide another generic and conceptually clear electronic setting for realizing NH topological phases (see Bergholtz and Budich, 2019 for a detailed discussion). In fact, the well-established theory of quantum transport that has been used and experimentally tested over decades of intense research is entirely based on NH physics (see, e.g., Datta, 2005). The more recent development is essentially the perspective that these problems can be recast in the systematic context of NH topology, which has already inspired suggestions for novel phenomena in experimentally accessible solid state setups. Let us consider such a setup, where one side of the junction is considered to be a thermal reservoir (lead), which induces a self-energy on the surface of the system thus leading to the effective NH system Hamiltonian
(31) |
where is the Hermitian Hamiltonian of the isolated system, and denotes the retarded self-energy at energies close to the chemical potential reflecting the coupling to the lead. Due to causality all eigenvalues of reside in the lower half-plane . Since is generically non-Hermitian and matrix-valued, it can have drastic implications for the topology of the interface states. This has been investigated in the context of superconducting junctions featuring EPs Pikulin and Nazarov (2012, 2013); Avila et al. (2019); San-Jose et al. (2016) as well as in interfaces between topological insulators coupled to ferromagnetic leads Philip et al. (2018); Chen and Zhai (2018); Bergholtz and Budich (2019). In the latter case, the Hall conductance in the gapped phase loses its quantization Philip et al. (2018); Chen and Zhai (2018) thus signalling a breakdown of the topological nature of the system well known from the Hermitian limit. Remarkably, however, the non-Hermiticity of this setup can also promote the topological properties: While the ferromagnet breaks time-reversal symmetry one would expect it to generally open a gap in the surface theory. As shown by Bergholtz and Budich, 2019, there is a critical angle of the magnetization beyond which the dissipation overcomes the gap, thus promoting the symmetry-protected surface topology to a nodal NH topological phase with EPs and NH Fermi arcs that does not rely on any symmetry.
Photonic and hybrid systems feature NH topology also in the quantum regime. A spectacular example of this is the concept of topological lasers Bahari et al. (2017); St-Jean et al. (2017); Zhao et al. (2018); Parto et al. (2018); Bandres et al. (2018); Harari et al. (2018); Longhi and Feng (2018); Longhi (2018). Lasers fundamentally depend on gain and the basic idea of topological lasers thus includes ingredients of topology, quantum mechanics and non-Hermiticity.
NH topology may also appear in less obvious ways as exemplified in the bosonic Bogoliubov-de Gennes (BdG) problem, which occurs naturally in various settings ranging from photons under parametric driving to exciton polariton systems Bardyn et al. (2016) and cold atomic gases Barnett (2013). Although superficially identical to the fermionic BdG problem well know from the theory of superconductivity, the transformation needed to diagonalize the BdG Hamiltonian for bosons is paraunitary rather than unitary and the corresponding spectra are not generally real. Indeed, parametric instabilities corresponding to complex eigenvalues are know to occur in several experimentally relevant settings Barnett (2013); Shi et al. (2017); Peano et al. (2016a, b); Galilo et al. (2015). As such these provide a distinct raison d’être for NH classification schemes as observed by Lieu, 2018b.
iv.2.2 Emergent dissipation in closed systems
At a global level, a closed quantum mechanical system undergoing unitary time evolution does not feature dissipation. However, local observables in interacting quantum many-body systems obey non-linear equations of motion thus effectively leading to dissipative dynamics. In this context, it has been proposed that dissipation in the form of emergent non-Hermiticity can have a profound impact on the low-energy description of interacting and disordered quantum matter Kozii and Fu (2017); Yoshida et al. (2018); Zyuzin and Zyuzin (2018). Phenomenologically, this scenario is reminiscent of the concept of eigenstate thermalization Deutsch (1991); Srednicki (1994), a generic feature of non-integrable quantum systems with a large number of degrees of freedom, where the system acts as its own thermal bath for local observables. In the present context, quasiparticles with a given momentum scatter off each other or at impurities and thereby acquire a finite life-time. The corresponding self-energy is non-Hermitian and, when sufficiently generic, one may thus expect it to feature, e.g., exceptional degeneracies and their concomitant phenomenology as discussed in Section II.1.2.
Along these lines, intriguing suggestions about emergent topological NH phenomena have been put forward in heavy fermion systems, which are natural due to the extreme renormalization of the bare electron properties Yoshida et al. (2018), in nodal semimetals, which, according to the general discussion in Section II.2.1, provide an ideal setting for NH nodal phases Zyuzin and Simon (2019); Moors et al. (2019); Zyuzin and Zyuzin (2018); Kimura et al. (2019); Yoshida et al. (2019b), in strongly correlated Kondo materials Michishita et al. (2019), and for magnons—the spin-wave excitations of quantum magnets—, which provide another very natural platform for NH topology as explored by McClarty and Rau, 2019. Bosonic BdG Hamiltonians also occur in the context of magnons, which provides an alternative way of arriving at NH phenomenology, e.g., in ferromagnetic materials Shindou et al. (2013) along the lines discussed in the context of open systems above.
Related ideas of emergent EPs have also been put forward early on in the context of nodal-line semimetals in the presence of an external magnetic field Molina and González (2018) and radiated by circularly polarized light González and Molina (2017). Furthermore, the interplay between non-Hermiticity and superconductivity at the level of toy models has been investigated Ghatak and Das (2018). Finally, we note that even when starting from entirely Hermitian systems, physical insights can be gained by formally extending a given model into the NH realm as has been shown for Majorana wires Mandal (2015) and interacting spin systems Luitz and Piazza (2019).
V Concluding remarks
To summarize, bringing together insights from recent literature, in this review article we have discussed how relinquishing the assumption of Hermiticity qualitatively modifies and enriches the notion of topological band structures. Both novel NH topological phases and fundamental changes to the bulk-boundary correspondence have been shown to be intimately related to the occurrence of exceptional degeneracies, a property unique to the complex spectra of NH matrices. These insights demonstrate that effective NH Hamiltonian approaches can, despite their appealing conceptual simplicity, describe intriguing topological phenomena relating to the presence of dissipation in both classical and quantum systems. This is in line with earlier findings in the fully microscopic context of quantum master equations that dissipation may be harnessed for the formation of ordered states of matter Diehl et al. (2008); Verstraete et al. (2009); Diehl et al. (2011), and is thus better than its destructive reputation. Despite the impressive recent progress, many open questions remain in the rapidly evolving field of NH topological matter. We would like to close our discussion by pointing out a few possible future perspectives.
Owing to the broad variety of experimental platforms for NH topological systems (see Section IV), a natural quest is to identify and experimentally implement potential technological applications of topological robustness and quantization in dissipative systems. Directly harnessing the analytical properties of exceptional degeneracies to enhance the sensitivity of a particle detector, Chen et al., 2017; Hodaei et al., 2017 reported on a promising steps in this direction. Topological lasers based on robust NH boundary and interface states Bahari et al. (2017); Bandres et al. (2018); Harari et al. (2018); St-Jean et al. (2017); Zhao et al. (2018); Parto et al. (2018) provide another exciting path toward potential new technology.
While the NH description of classical systems is quite satisfactorily understood within a single-particle or wave picture, the conceptually more complex case of open quantum many-body systems effectively described by an NH Hamiltonian is still far from a conclusive description. A few natural open questions in this context include: The precise relation between different levels of description, ranging from exact Liouvillian quantum dynamics to effective NH Hamiltonians, in particular in the context of topological properties. New topological phases beyond the independent particle picture: While intriguing NH effects in interacting systems have been reported Roncaglia et al. (2010); Yoshida et al. (2019a); Carlström (2019); Lee et al. (2019c); Matsumoto et al. (2019); Luitz and Piazza (2019); Mu et al. (2019), qualitatively new fractional topological phases that may be seen as a genuinely NH counterparts to fractional quantum Hall states or spin liquids familiar from strongly correlated Hermitian systems are yet to be discovered.
Acknowledgements.
We would like to thank Johan Carlström, Vatsal Dwivedi, Elisabet Edvardsson, Loïc Herviou, Kohei Kawabata, Rebekka Koch, Ching Hua Lee, Simon Lieu, David Luitz, Francesco Piazza, Stefano Longhi, Bernd Rosenow, Henning Schomerus, Marcus Stålhammar, Kang Yang, and Tsuneya Yoshida for discussions. E.J.B. and F.K.K were supported by the Swedish Research Council (VR) and the Wallenberg Academy Fellows program of the Knut and Alice Wallenberg Foundation. F.K.K. was also supported by the Max Planck Institute of Quantum Optics (MPQ) and Max-Planck-Harvard Research Center for Quantum Optics (MPHQ). J.C.B. acknowledges financial support from the German Research Foundation (DFG) through the Collaborative Research Centre SFB 1143 (Project No. 247310070) and the Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter ct.qmat (EXC 2147, ProjectNo. 39085490).References
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