Crossing exceptional points without phase transition

We show that the theoretical framework linking exceptional points (EPs) to phase transitions in parity-time (PT) symmetric Hamiltonians is incomplete. Particularly, we demonstrate that the application of the squaring operator to a Jx PT lattice dramatically alter the topology of its Riemann surface, eventually resulting in a system that can cross an EP without undergoing a symmetry breaking. We elucidate on these rather surprising results by invoking the notion of phase diagrams in higher dimensional parameter space. Within this perspective, the canonical PT symmetry breaking paradigm arises only along certainprojections of the Riemann surface in the parameter space.


Crossing exceptional points without phase transition Qi Zhong & Ramy El-Ganainy
We show that the theoretical framework linking exceptional points (EPs) to phase transitions in paritytime (PT) symmetric Hamiltonians is incomplete. Particularly, we demonstrate that the application of the squaring operator to a Jx PT lattice dramatically alter the topology of its Riemann surface, eventually resulting in a system that can cross an EP without undergoing a symmetry breaking. We elucidate on these rather surprising results by invoking the notion of phase diagrams in higher dimensional parameter space. Within this perspective, the canonical PT symmetry breaking paradigm arises only along certainprojections of the Riemann surface in the parameter space.
Exceptional points (EPs) are peculiar singularities associated with multivalued complex function 1 . They also arise as special degeneracies in the spectra of parity-time-(PT-) symmetric (and in general non-Hermitian) Hamiltonians at which the eigenvalues and the corresponding eigenvectors become identical 2,3 , thus signaling a collapse of the eigenspace dimensionality, which in turn gives rise to a host of intriguing effects in the vicinity of these singularities 4,5 . One particular characteristic feature that has been studied thoroughly in literature is phase transitions across EPs in PT-symmetric arrangements 6 . When the symmetry of the eigenvectors are studied as a function of one parameter of the PT Hamiltonian, one finds that on one side of the EP, the eigenvectors respect PT symmetry (i.e. they commute with the PT-symmetric operators) whereas on the other side of the EP, they violate the PT symmetry (in fact applying PT operator to one eigenvector yields another eigenvector). This phase transition between the PT phase and broken PT (BPT) phase, which is also known as PT spontaneous symmetry breaking, is accompanied by complex eigenvalue bifurcation. This behavior has been experimentally demonstrated in various physical platforms in optics 7-10 , electronics 11 and acoustics 12 . For more information, see refs 13,14 .
In light of these intense theoretical and experimental activities, it is perhaps not surprising that EPs are always associated PT phase transitions. What is surprising though, is the lack of any rigorous mathematical proof for this statement. In this work, we show that this is not a coincidence, and that this widely accepted picture of PT phase transition is in fact incomplete.
Before we proceed, we briefly review the archetypal discrete PT-symmetric Hamiltonian, which was the subject of detailed investigation in several studies 7,9,10,15,16 . It consists of two coupled elements, having coupling coefficient κ and balanced gain/loss profile characterized by the non-Hermitian parameter γ. This Hamiltonian respects PT symmetry in the κ − γ plane. However, as γ is varied from γ < κ to γ > κ, the associated eigenvectors undergo a spontaneous symmetry breaking from the PT phase to the BPT phase 13,14 . The transition point separating these two phases (γ = κ) is an EP. This behavior, which we call canonical PT phase transition, has been reported in more complex discrete and continuous systems. This in turn led to the common belief that crossing EPs along straight lines and PT phase transitions are inseparable notions.
In this work, we show that this is not the whole story and that this picture is indeed incomplete. To do so, we use the squaring operator to construct a simple Hamiltonian that violates the canonical PT phase transition in the following sense: as one parameter is varied continuously and monotonically along a straight line, the system crosses an EP without any PT symmetry breaking. As we will shortly see, this is an outcome of the non-trivial topological features incurred on the Riemann surface by the squaring operation. For more detailed discussion on how square and square root operations can give rise to altogether new topological structures, see ref. 17 .
To this end, consider the following family of PT Hamiltonians H M (which can be generated by using the recursive bosonic algebra method 18 ) whose matrix elements are given by: where γ and κ are the gain (or loss) and coupling coefficients, respectively and = SCIenTIfIC REPORTs | (2019) 9:134 | DOi:10.1038/s41598-018-36701-9 The eigenvalues of H M are given by μ where m = 0, 1, 2, ‥‥, M 18 ; and they feature higher order EPs [19][20][21][22] at γ = κ with a phase transition across this point from PT phase (γ > κ) to the BPT phase (γ > κ) 18,23 . The crucial observation here is that the eigenvalues are pure real (imaginary) in the PT (BPT) phase.

Let us now consider a new Hamiltonian
. Furthermore, The eigenvalues of  are given by λ = μ 2 . The eigenspectrum of  M is thus composed of degenerate subspaces with positive real eigenvalues when γ < κ and negative real eigenvalues when γ > κ. A schematic representation of how this degeneracy arise is shown in Fig. 1. Clearly, the spectrum of  M does not undergo complex eigenvalue bifurcation as the parameter γ is swept across a straight line that passes through the point γ = κ. Additionally, an important consequence of the two-fold degeneracy of the spectrum of M  is that, it is always possible to construct PT symmetric eigenstates for any γ and κ which is not the case for H M (see Supplementary Note A for more detailed discussion). It thus appears that the point γ = κ is not associated with phase transition. Naturally, one would then ask if it is an EP. In order to check this and without in loss of generality, we consider the case where M = 3 (the Hamiltonian 3  has dimensions 4×4) and we set κ = 1. In general the Hamiltonian 3  has four different eigenstates forming two degenerate subspaces. Figure 2(a) depicts the Hermitian angle (Θ) between the two planes consisting of the degenerate eigenvectors (see Supplementary Note B for the definition of Hermitian angle between two planes). One indeed sees that at γ = 1, the two planes are identical (Θ = 0), which indicates a collapse of the eigenspace dimensionality as would be expected at an EP. Interestingly however, at this point  3 exhibit two different eigenvectors whereas H 3 has only one. We elaborate more on this feature in Supplementary Note A. Figure 2(b) and (c) plot the values of  λ Λ = − ′ − I ( ) 3 1 as a function of the parameter λ′ for two different cases when γ = 0.1 (away from the EP) and γ = 0.9 (close to the EP). The spread of the high values of Λ in the λ′ plane indicates the system sensitivity to perturbations (see Supplementary Note C for detailed definition of pseudospectrum method). As can be inferred from the figure, this is indeed the case in the vicinity of point γ = 1. Based on the above analysis, we indeed conclude that, as the Hamiltonian  3 is swept across the straight line passing through the point γ = 1, it crosses an EP without experiencing a phase transition. The above example thus presents a very interesting scenario that demonstrates the possibility of violating the canonical PT symmetry breaking (in Supplementary Note D we show that this example is not unique).
In order to better understand these results within a general framework that encompass all the possible different situations, we invoke the notion of phase diagrams. In statistical mechanics and nonlinear dynamics, phase diagrams are used to classify the system's behavior into different phases as a function of some external parameters. In the context of our discussion of 3  , one should in general study the classification of the eigenstates as a function of 32 different parameters (16 complex matrix entries). Fortunately, we can gain an insight into the behavior of 3  by considering a low dimensional projection of this higher dimensional parameter space. Here we do so by fixing κ and de-correlate some of the other parameters by allowing (1, 1) and  (4, 4) 3 to vary as a function of γ 1 while the rest of the matrix elements vary with γ 2 . This choice provides a 2D projection of the phase diagram while at the same time guarantees that 3  still respect PT symmetry. Figure 3(a) and (b) depict the Riemann surfaces for the real and imaginary components of the eigenvalues of 3  in the γ 1 -γ 2 plane, where one can identify the distinct PT phases. Figure 3(c) plots the phase diagram as extracted from Fig. 3(a). Figure 3(d) presents a more detailed blow up of the area surrounded by the rectangle in (c). Figure 3(d) clearly demonstrates the different phases are separated by curved lines of EPs. As one varies one or more of the system's parameter, the behavior can be very different depending on the trajectory taken in the parameter space. For example, no phase transition is observed if γ 1 is varied while γ 2 = 0. On the other hand, fixing γ 2 = 1 and sweeping γ 1 from negative to positive values will lead to a PT-BPT phase transition followed by a BPT-PT transition (blue line). One can also fix γ = − ≈ . 100 12 69 0 566 2 to a value that guarantees that the line swept by varying γ 1 will just touch one EP without any phase transition (green line touching EP 2 ). The particular case we discussed earlier for  3 corresponds to the line γ 1 = γ 2 which also touches the boundary at one EP (red line crossing EP 1 ) without any phase transition. The concept of higher dimensional phase diagram thus provides a unified umbrella to treat all the rather special cases of phase transition, reverse phase transition, EP without phase transition, no EP and no phase transition as well as multiple phase transitions. This generalized perspective is very important to complex non-Hermitian systems and design next generation experiments. As a side note, we emphasize that the exceptional lines separating the different phases in Fig. 3 curves in the parameter space, which are very different from previous studies that demonstrated exceptional lines in the Fourier space 24,25 .
In summary, we have revisited the concept of PT phase transition across EPs and demonstrated that, contrary to the common belief, crossing an EP along straight lines (the case of curved trajectories is rather trivial) in the parameter space can take place without PT spontaneous symmetry breaking. We have explained these results by introducing the concept of PT phase diagram and its different projections in the parameter space that characterize a PT symmetric Hamiltonian. Our work provides a new twist and a deeper understanding for the physics of non-Hermitian systems near EPs, with potential implication in various fields such as photonics 14 , acoustics 12,26 and electronics 11,27,28 . Particularly, the recent important work on driving Floquet PT symmetric systems offer a natural platform for confirming our predictions experimentally 29 . Our results also raise interesting questions about the evolution along closed loops [30][31][32][33][34][35][36][37] in these higher dimensional parameter spaces, which we plane to investigate elsewhere. Finally it is instructive to compare the behavior discovered here in this work with other systems studied in condensed matter physics. For example, by referring to the phase diagram of the quantum phase transition associated with the Bose-Hubbard Hamiltonian of interacting chain [38][39][40] (Fig. 4), we can see that it is possible to choose a trajectory that crosses the critical point on the boundary between the superfluid (SF) and Mott insulator (MI) phases from the SF side without having a phase transition.