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Fate of exceptional points in the presence of nonlinearities

The non-Hermitian dynamics of open systems deal with how intricate coherent effects of a closed system intertwine with the impact of coupling to an environment. The system-environment dynamics can then lead to so-called exceptional points, which are the open-system marker of phase transitions, i.e., the closing of spectral gaps in the complex spectrum. Even in the ubiquitous example of the damped harmonic oscillator, the dissipative environment can lead to an exceptional point, separating between under-damped and over-damped dynamics at a point of critical damping. Here, we examine the fate of this exceptional point in the presence of strong correlations, i.e., for a nonlinear oscillator. By employing a functional renormalization group approach, we identify non-perturbative regimes of this model where the nonlinearity makes the system more robust against the influence of dissipation and can remove the exceptional point altogether. The melting of the exceptional point occurs above a critical nonlinearity threshold. Interestingly, the exceptional point melts faster with increasing temperatures, showing a surprising flow to coherent dynamics when coupled to a warm environment.

Introduction -Quantum mechanics and its Hamiltonian description focus on a closed system premise, exhibiting coherence that is the key resource for unlocking future quantum technologies [1,2]. In realistic situations, however, any quantum system is coupled to an environment that results in decoherence and dissipation [3], which generically poses an obstruction for quantum applications. At the same time, exploiting dissipation can result in a variety of advantageous novel effects, such as engineering of robust quantum communication [4], dissipation-induced topological effects [5], and dissipative phase transitions [6][7][8][9].
To address such open quantum systems, Liouvillians are routinely used [10][11][12]. Whereas closed systems have a real spectrum and a corresponding unitary time evolution, open systems are generally described by a complex spectrum [13][14][15] and evolve with non-Hermitian operators. Similarly to phase transitions in closed systems, the open system exhibits critical points in parameter space, where the gap in the complex spectrum closes. Such critical points, which correspond to degeneracies in the open system, are known as exceptional points (EPs) [16], with the most ubiquitous EP example manifesting for the damped harmonic oscillator (DHO) at critical damping [17]. In recent years, EPs have gained considerable attention due to the variety of the applications that they offer, e.g., in quantum sensing [18,19], in photonics [20,21], and in topology [22].
In both closed and open systems, nonlinearities result in collective many-body effects that are oftentimes challenging to describe theoretically. They offer a plethora of rich strongly-correlated physics ranging from Kondo effects [23,24] to superfluidity [25,26], and to hightemperature superconductivity [27]. One of the paradigmatic examples to see such many-body effects is the φ 4 theory [28]; the building block of quantum field theory [29] with implications across many fields, ranging  1)]. The quadratic (black) potential is corrected by repulsive (α > 0) quartic nonlinearities (red lines depict different α). The particle's motion is damped by the environment (blue bath with rate γ, wiggly blue arrow). (b) The real and imaginary parts of the characteristic exponent as a function of the oscillator's Q-factor, ω0 γ. The real part of in the under-damped region (red background) has two branches that merge together at an exceptional point, i.e., at critical damping γ = 2ω0. In the over-damped case (blue background), the real part of vanishes. The background color depicts the order parameter ∆.
from superfluidity [30] to high-energy physics [31]. To study a φ 4 theory in a closed system, renormalization group approaches are commonly employed [32,33] as conventional perturbative treatments break down [34,35]. In open systems, the competition between coherent processes and the incoherent forcing induced by the environment raises a fundamental question regarding the robustness of collective many-body effects against dissipation. Hence, the study of strong correlations in dissipative quantum systems is drawing much contemporary interest [36][37][38][39], albeit with a focus on how the nonlinearity acts as the source for the non-Hermitian system. arXiv:2208.11205v1 [cond-mat.str-el] 23 Aug 2022 In this work, we explore the fate of exceptional points in nonlinear systems. We introduce nonlinearity as a quartic correction to the potential energy of a DHO, and find that the φ 4 theory washes away the dissipationinduced EP above a critical threshold. This implies that the impact of a dissipative environment can be suppressed with many-body interactions. Furthermore, we study the influence of thermal fluctuations in the system, and find that they amplify the impact of the nonlinearity. Thus, the EP can be removed by heating the system. We propose a scheme by which to measure our prediction in a plethora of systems, and expect that our result bears implications to a broad range of fields.
Model -We study a damped nonlinear oscillator (DNO) governed by the equation of motion where q is the displacement, dots mark differentiation with respect to time, ω 0 is the bare frequency of the harmonic oscillator, γ is the linear dissipation rate, and α is the Duffing nonlinearity. Equation (1) describes many physical systems, ranging from mechanical, electrical, and optical resonators [40,41] to the inflation of the universe [31]. As an example, we consider a massive particle oscillating in a quartic 1D potential energy while dissipating energy to the walls, see Fig. 1(a).
In the presence of the Duffing term, α ≠ 0, the nonlinear Eq. (1) cannot be solved analytically for generic parameters [41]. Instead, various approximate methods are employed. For example, deep in the underdamped regime and for small α, perturbative approaches such as the Poincaré-Lindstedt [42] and Green's functions diagrammatics [41] were developed. In this work, we go beyond previous studies by considering the whole crossover from under-damped to over-damped regimes using a full quantum mechanical treatment. In particular, we address the impact of a small nonlinearity, α max{ω 0 , γ} 3 ≪ 1, on the exceptional point appearing for the quantum DHO. To this end, we formulate the problem in the imaginary-time formalism [43], [44] where the action for the DNO takes the form where β is the inverse temperature, φ ( * ) is the quantum field operators corresponding to the position operator, and the free imaginary-time Green's function is defined as the solution to the equation of motion corresponding to the first three terms in Eq. (1), namely, it satisfies . In Matsubara space, G(iω n ) = 1 β ∫ β 0 dτ G(τ )e iωnτ , and the bare Green's function reads where ω n = 2πn β denotes the bosonic Matsubara frequencies with n ∈ N. Using analytic continuation, we can define the retarded free Green's function G R 0 (ω) = G 0 (iω n → ω + i0 + ), where the real and imaginary parts of its poles determine the energy and lifetime of excitations in the system, respectively. Analogous to our above discussion [cf. Fig. 1], we identify the under-damped region, γ < 2ω 0 , for which G R 0 (ω) contains poles that have both a real (frequency), Re{ } = ± ω 2 0 − (γ 2) 2 , and an imaginary (lifetime) part, Im{ } = γ 2. The under-damped region ends at the critically-damped point, γ = 2ω 0 , for which the real part vanishes, i.e., at the exceptional point of this model.
Perturbation theory -To answer how the nonlinearity impacts the existence of exceptional points, we focus on the dressed imaginary-time Green's function, (2)]. Using a perturbation theory (PT) approach and employing Dyson's equation, we can write the dressed Green's function as where we defined the selfenergy Σ in terms of a perturbative expansion in the nonlinearity strength as [45] Our treatment can be readily written in terms of Feynman diagrams [46], where we focus on the one loop correction (Hartree term). Plugging Eq. (3) into Eq. (4), we first focus on the zero temperature limit, where the self-energy reads [46] for the different regions of the model. It is important to note that in the over-damped regime, the perturbative self-energy diverges logarithmically for ω 0 ≪ γ.
We obtain a dressed retarded Green's function PT ] with the renormalized oscillator frequency defined as ω PT = ω 2 0 + Σ PT . Such a loop correction arises purely from quantum fluctuations of the oscillator. For α > 0, the renormalized frequency is always larger than the bare one, ω PT > ω 0 [cf. Eq. (5)]. In other words, the nonlinearity increases the frequency of oscillations in the system. Such a renormalization of the oscillation frequency can be understood intuitively: the positive nonlinearity increases the restoring force, i.e., it makes the potential energy steeper as a function of position, see Fig. 1(a). Crucially, for sufficiently large α, we can have a situation where ω 0 < γ 2 but ω PT > γ 2, such that the EP of the system is thus lifted, and the over-damped phase cannot be realized, see Fig. 2(a). For a small nonlinearity, instead, the EP manifests albeit at a smaller Q-factor as compared to the linear case, see Fig. 2(b). This implies a critical value of nonlinearity α crit beyond which the exceptional point vanishes.
The perturbative results predict a non-monotonic behaviour of the excitation energy Re{ } as a function of ω 0 [cf. Eq. (5) and Fig. 2], which implies a fragility of EPs to nonlinearities. Our result however is not physical, e.g., by decreasing the bare oscillation frequency, the system can go from over-damped to under-damped dynamics, see Fig. 2(b). Such unphysical behaviour combined with the logarithic divergence in the perturbative expansion [cf. Eq. (5)], raise questions regarding the reliability of the PT and its prediction concerning EPs. This motivates applying a non-perturbative approaches to reveal the true impact of nonlinearities on EPs, which we perform in the following.
Functional renormalization group -To cure the above logarithmic divergency, we apply a functional renormalization group approach, in which a flow parameter Λ ∈ [0, ∞] is introduced in the free Green's function G Λ 0 , and high-frequency degrees of freedom (compared to Λ) are integrated out [47,48]. Following this procedure, we are able to approach the over-damped region with care, and obtain a hierarchy of differential equations (flow equations) for the one-particle irreducible vertex functions, e.g., for the self-energy Σ Λ [46]. Whereas this procedure is formally exact, we need to truncate the hierarchy of flow equations in order to have a tangible solution. Here, we use the so-called first-order truncation scheme, which is controlled for weak nonlinearities, such that only the self-energy flows as with the scale-dependent Green's function To compare with the perturbative result in Eq. (5), we first focus on the zero temperature limit of the flow (S30). To this end, we consider the so-called sharp cutoff scheme, where the free Green's function is scaled as We solve this flow numerically and denote the solution Σ FRG . In Fig. 3(a), we plot Σ FRG and compare the result with the perturbative treatment [cf. Eq. (5)]. Crucially, we observe that the logarithmic divergence is regularized in the FRG approach. Instead, the self-energy saturates to a finite value for ω 0 ≪ γ. We proceed and insert the FRG self-energy into the retarded Green's function,  Fig. 1(a). Increasing the temperature, expands the phase space that the trapped particle explores in the potential well (red fill and arrow). (b) The FRG resulting phase diagram [cf. Eqs. (S30) and (8)] as a function of temperature and Q-factor with γ = 1 and α = 0.2 [cf. horizontal line in Fig. 3(b)]. (c) Same as (b), as a function of temperature and nonlinearity for γ = 1 and ω0 = 0.1 [cf. vertical line in Fig. 3(b)]. To do: better resolution plots! and obtain a corresponding regularized-renormalization of the oscillator frequency ω FRG ≡ ω 2 0 + Σ FRG . Similarly to the results presented in Fig. 2, we systematically study the fate of the EP as a function of α and ω 0 γ. In Fig. 3(b), we collect the outcome into a phase diagram. Importantly, the unphysical phase predicted by the PT at ω 0 ≪ γ vanishes [cf. Fig. 2(b)].
We can now draw a conclusion concerning the fate of the critical-damping EP. At weak nonlinearity, the exceptional point is pushed to smaller ω 0 γ, and the overdamped region is shrinking. This fits with our intuition that the effective oscillation frequency grow with increasing α, and counters the dissipation. Interestingly, beyond a critical nonlinearity value α crit , the EP is fully removed from the system. Physically, beyond the critical value, dissipation cannot counter the coherent forcing enacted by the quartic potential. This is the main result of our paper.
Finite Temperature -Our treatment thus far dealt with the fate of the EP at zero temperature, T = 0. With increasing temperature, thermal fluctuations excite the DNO, i.e., they allow the trapped particle in the potential well to explore positions further away from the bottom of the well, see Fig. 4(a). Fortunately, our Matsubara formalism in Eqs. (4) and (S30) allows for the exploration of the DNO's EP physics at finite temperatures. From Eq. (4), we observe that also at T ≠ 0, the PT yields in unphysical divergences [46], which we want to resolve with the aid of the FRG approach.
To set up FRG for T > 0, we modify the sharp cutoff with a smeared one [49], i.e., the scale-dependent Green's function now reads G Λ 0 (iω n ) = G 0 (iω n )Θ T ( ω n − Λ) with smeared cutoff With this cutoff inserted into Eq. (S30), we numerically solve the resulting truncated flow equation [46] and obtain a phase diagrams akin to that of Fig. 3(b), see Figs. 4(b) and (c). We observe that with increasing temperatures, the over-damped regions shrink, and specifically that α crit goes down with temperature T . This is a result of an interesting interplay between thermal fluctuations and the impact of the nonlinearity on the system. Specifically, we can understand this interplay using the following semi-classical picture: thermal fluctuations kick the particle away from the basin where the quadratic (harmonic) approximation holds, i.e., it explores large displacements where nonlinearities are more pronounced.
As the nonlinearity α > 0 increases the renormalized oscillator frequency and thus reduces the over-damped region, we observe the melting of the over-damped region with temperature, see Fig. 4(b). This results in an exotic case where rising temperatures lead to more coherent motion in the system by enhancing the impact of the nonlinearity.
Exceptional points mark observable criticality in open systems.
We predict an exotic interplay between such environment-induced phenomenon and the intrinsic many-body interactions in the closed system. A possible route to measuring our predictions involves starting in an over-damped system, Q < 1 2, and tuning the nonlinearity of the device or by increasing the temperature, see

A. Action of a damped harmonic oscillator
For completeness, we provide here the derivation of the imaginary-time action of a damped harmonic oscillator (DHO) [1]. We begin with the action of a harmonic oscillator (HO) whose position satisfies the equation of motion q+ω 2 0 q = 0, or equivalently whose Hamiltonian reads H 0 =q 2 2 +ω 2 0 q 2 . The partition function of the HO is Z = Tr{e −βH0 }, which can be formulated in the path-integral formalism as Z = ∫ Dx e −S0 , with the imaginary-time action where the oscillator Green's function g 0 satisfies We now investigate how such a Green's function gets modified if the HO is in contact with a heat bath. The latter has a dispersion Ω k , composed of infinitely-many oscillators with the action where g b is the bath's Green's function, satisfying −∂ 2 and the action corresponding to the coupling of the HO to the bath reads with λ k being the coupling constant to the k − th oscillator in the bath. We can integrate out the bath's degrees of freedom and obtain with and the bath-induced self-energy We can identify by plugging Eq. (S4) in Eq. (S8). Defining the bath spectral density as [1] we can rewrite the bath-induced self-energy as (S11) Furthermore considering an Ohmic bath with band-width ω c , the bath spectral density reads and Eq. (S11) becomes corresponding to Ohmic dissipation.

B. Action of a damped harmonic oscillator in Keldysh formalism
Commonly, an open system scenario is better treated using a Keldysh action formalism [2]. We present here, for completeness, that in our case of a single oscillator coupled to a single Ohmic bath, the Matsubara description suffices [2]. The bath-induced self-energy in the keldysh formalism reads where the bath's Green's function are , and (S15) We can identify that the fluctuation-dissipation theorem holds for the bath, i.e., with n b (ω) = 1 [e βω − 1] being the bosonic distribution function, and β the inverse temperature. Plugging Eq. (S16) in Eq. (S14), the Keldysh component of the bath-induced self-energy becomes with the definition and the retartded/advanced bath-induced self-energy reads (S20) If we consider an Ohmic spectral density of the form Eq. (S12), we obtain Note that in the limit of ω c → 0, we recover the fluctuation-dissipation relation for the bath-induced self-energy

C. Perturbation theory for DNO
In this section, we outline the calculation of the self-energy for a nonlinear damped Oscillator (DNO) within first order perturbation theory, i.e., one loop correction (Hartree term), as it is shown diagramatically in Fig. 5(a), and written in equation form as where G 0 is the free Green's function, L 2 is the two-particle vertex function, and n = 1, 1 ′ , 2, 2 ′ enumerates the degrees of freedom. For the DNO, G 0 (n; n ′ ) in (Matsubara) frequency space depends on a single frequency, and the vertex function is frequency independent, therefore, we can simply replace L 2 by nonlinearity α, and obtain Eq. main text] results in with ω n = 2nπ β, n ∈ N, τ > 0 a convergence factor [3], and Ω 0 = ω 2 0 − (γ 2) 2 , which is real in the under-damped case and purely imaginary in the over-damped region. We emphasize that this equation is valid for arbitrary temperatures T , and all the information about the temperature dependence of the self-energy is encoded in the Matsubara summation over n in Eq. (S24). In order to evaluate such a summation, we rewrite the Matsubara sums in terms of contour integrals [3]. Specifically, we can show ∫ dz 2πi as the integral over the semi-circle with radius R → ∞ vanishes as τ ∈ (0, β), and hence n b (z)e τ z → 0 for Re(z) → −∞. Furthermore, using the Residue theorem, we can rewrite Therefore the perturbative self-energy of Eq. (S24) becomes ] as a function of temperature for various oscillator frequencies. The inset shows that the self-energy depends linearly on temperature for T ≫ ω0.

Zero temperature limit
In the limit of zero temperature, we can approximate the bosonic distribution function as lim β→∞ n b (ω) = −θ(−ω), and therefore the integral in Eq. (S28) can be evaluated as which simplifies to Eq. (5) in the main text for different regions of the DNO.

Finite temperatures
At finite temperatures, we can evaluate the integral in Eq. (S28) numerically, see outcome in Fig. 6. The renormalized oscillator frequency is independent of T at T ≪ ω 0 , i.e., when thermal energy is not enough to excite the oscillator. However, at T ≫ ω 0 , we see a linear dependence on temperature. At such high temperatures, we can approximate the bosonic distribution function in Eq. (S28) as lim β→0 n b (ω) = 1 (βω), which explains the linear dependence of the self-energy on temperature, see inset to Fig.6. Note that the perturbative self-energy diverges at ω 0 → T , as it is shown in Fig. 6. (b).

D. FRG for DNO
In this section, we elaborate on the derivation of FRG flow equations for the DNO and their numerical solutions presented in the main text. In the FRG approach [4], a scale parameter Λ ∈ [0, ∞] is introduced. The corresponding scale-dependant free propagator G Λ 0 is defined such that it vanishes in the beginning of the flow G Λ→∞ 0 = 0, and it approaches the bare Green's function at the end of the flow G Λ→0 0 = G 0 . Generally, the flow equation for the self-energy reads where S Λ is the single-scale propagator, defined as with G Λ = [(G Λ 0 ) −1 − Σ Λ ] −1 and L Λ 2 is the two-particle vertex function from Eq. (S30), which generically has its own flow equation, see Ref. 4. We focus here on the first-order truncation scheme, where the two-particle vertex function stays the same during the flow, i.e., L Λ 2 ≡ L 2 and is frequency-independent. Hence, Eq. (S30) simplifies to Eq. (6) in the main text, which dictate the flow of the self-energy at any temperature, provided that the scale-dependant free propagator G Λ 0 is chosen appropriately.

The zero temperature limit
At T = 0, the distance between Matsubara frequencies vanishes and we can treat them as a continuous variable ω n → ω. In this limit, we work with the sharp cut-off scheme, where G Λ 0 = θ( ω −Λ)G 0 , and thereby Eq. (S31) becomes Furthermore, employing the Morris Lemma [5], we obtain which simplifies Eq. (6) of the main text to It is important to note that if we neglect the self-energy in the right-hand side of the flow Eq. (S34), and integrate from Λ = ∞ to Λ = 0, we re-obtain the perturbative results Eq. (S24), when the Matsubara sum can be replaced by an integral in the zero temperature limit.

Finite temperatures
At finite temperatures, following the procedure introduced in Ref. 6, we introduce the scale-dependent Green's function G Λ 0 (iω n ) = G 0 (iω n )Θ T ( ω n − Λ), with the smeared cut-off defined in Eq. (8) in the main text. Therefore, the single-scale propagator As the Matsubara frequencies have spacing of 2π β, at a given scale Λ, there is only one m ∈ Z, for which ω m lies in the interval [−πT − Λ, πT − λ]. This removes the summation in Eq. (S30), and the flow equation for self-energy becomes which can be integrated with standard routines. Similar to the zero temperature limit, we can re-obtain the perturbative results by neglecting the self-energy on the right-hand side of Eq. (S36), as is shown in Fig. 7(a). The inclusion of the self-energy, however, will regularize the logarithmic divergencies appearing in the perturbative expansion, see Fig. 7(b). The renormalized self-energy is independent of temperature for T ≪ ω 0 , similar to the PT case.