The Physics of Spontaneous Parity-Time Symmetry Breaking in the Kelvin-Helmholtz Instability

We show that the dynamics, in particular the Kelvin-Helmholtz (KH) instability, of an inviscid fluid with velocity shear admits Parity-Time (PT) symmetry, which provides a physical explanation to the well-known observation that the spectrum of the perturbation eigenmodes of the system is symmetric with respect to the real axis. It is found that the KH instability is triggered when and only when the PT symmetry is spontaneously broken. The analysis of PT symmetry also reveals that the relative phase between parallel velocity and pressure perturbations needs to be locked at $\pi/2$ when the instability is suppressed.

the fluid equations, ∂ t ρ + ∇ · (ρv) = 0, (1) where ρ and p are the density and pressure of the fluid, v = (v x , v z ) is the velocity field in the (x, z) plane, and γ is the ratio of specific heats.
Assuming (v 1x , v 1z , p 1 ) ∼ exp[i(kx − ωt)] for a wavelength k ∈ R, the linearized equations can be written as where denotes d/dz, and variables However, the Hamiltonian H here is not Hermitian. As we will see, H is PT-symmetric instead.
Now we develop the mathematical tools for analyzing the physics of PT symmetry contained in Eq. (6). In general, a Hamiltonian H is PT-symmetric if where P is a linear operator satisfying P 2 = 1, and T is the complex conjugate operator [21].
In previous studies, two types of PT-symmetric Hamiltonians have been extensively studied.
The first type consists of scalar differential operators [1, 4,22,23], e.g., H = ∂ 2 x + (ix) N . The second type is linear maps of finite-dimensional complex vector spaces, which can be expressed as finite-dimensional square matrices [5,10,13,24], for example, The Hamiltonian operator (6) studied here represents a new type, which takes the form of matrix differential operators, i.e., matrices whose elements are differential operators. This type of Hamiltonians was encountered in the study of Bose-Einstein condensates [25], where Pauli matrices were used to describe the spin degree of freedom. An m-th order matrix differential operator can be written as where N i are n × n complex matrices. The corresponding state is ψ(z) = (ψ 1 , · · · , ψ n ) T .
This new type is a marriage between the two types of Hamiltonians described above. When  PT If the parity operator P is an n × n constant matrix in C n , then the PT operator commutes with the differential operators ∂ i /∂z i , and Eq. (11) can be simplified to This means that matrix differential operator H is PT-symmetric if all matrices N i are PTsymmetric with respect to the same PT operator. We will focus on this special case in the present study.
The most important and well-known properties of a PT-symmetric Hamiltonian H [21,26] can be summarized as follows. (i) Its spectrum is symmetric with respect to the real axis, i.e., if ω is an eigenvalue of H, so is its complex conjugateω. (ii) If every eigenfunction ψ of H is also an eigenfunction of the PT operator, i.e., PT ψ = λψ for some λ, then we say that PT symmetry is unbroken. In this case, the spectrum of H is always real.
Proof. First, select an arbitrary unit vector β 1 in the Hilbert space. If Aβ 1 = −β 1 , define a unit vector α 1 = c 1 (β 1 + Aβ 1 ), where c 1 is a real normalization constant. We have If Aβ 1 = −β 1 , define α 1 = iβ 1 and we have: Next, choose another unit vector in β 2 that is orthogonal to α 1 , i.e., β † 2 α 1 = 0, where † is the conjugate transpose operation. Construct a unit vector α 2 from β 2 using the same where use is made of the anti-unitarity of A in the third equal sign. If Aβ 2 = −β 2 then let α 2 = iβ 2 , and obviously α † 1 α 2 = 0. Thus, α 2 is orthogonal to α 1 . The amplitude of α 2 can be normalized to one by multiplying a real constant. Repeating the same procedure described above, we can build a set of complete orthonormal basis {α i } satisfying Aα i = α i .
in terms of one component φ l of the state vector φ, and the coefficients a i (ω) are real-value functions in the sense that a i (ω) ∈ R for ω ∈ R, or equivalently, Proof. Notice that PT is an anti-unitary operator in C n and (PT ) 2 = 1. According to Lemma 1, we can choose a set of complete orthonormal basis {X i } for C n satisfying PT X i = X i . Here, each X i is a constant column vector in C n and X † i X j = δ ij . Let O ≡ (X 1 , · · · , X n ) † , which belongs to U (n). Let the new state vector is φ = Oψ. In terms of φ, the Schrödinger equation is Following the procedure in [28], we can prove thatÑ i are real as follows. Because PT is anti-unitary and N i commutes with PT , we have This proves part (a). EquationHφ = ωφ consists of n coupled linear differential equations with dependent variables φ = (φ 1 , · · · , φ n ) T . In principle, we could eliminate n − 1 components of φ in favor of one φ l , and the resulting governing equation assumes the form of Eq. (16). SinceÑ i are all real matrices, the coefficient a i (ω) of the reduced ODE (16) must be real-value functions.
A corollary of Theorem 1 is that the coefficients of the characteristic polynomials of N i are real. When N i = 0 (i > 0), the Hamiltonian does not contain differential operators and can be represented by a complex matrix N 0 . In this special case, Theorem 1 implies that the coefficients of the characteristic polynomial of N 0 , which determines the spectrum of the system, are real [26].
We now return to the governing system for the KH instability (5). Its Hamiltonian (6) contains two components, It is easy to verify that N 0 and N 1 are PT-symmetric for P = diag(1, −1, 1), i.e., Thus, H is indeed PT-symmetric. It follows that the spectrum of the system is symmetric with respect to the real axis, and the KH instability is triggered when and only when PT and the transformed Hamiltonian is The matricesÑ 0 andÑ 1 are real, and the governing system can be reduced to a single ODE satisfying the coefficient condition (17). Indeed, from Eq. (5), it is straightforward to eliminate v 1x and p in favor of v 1z to obtain one single second-order ODE, Here, c = ω/k is the phase velocity. Obviously, condition (17) is satisfied for Eq. (26).
At the M → 0 limit, this equation becomes the Rayleigh stability equation. Due to the term (v 0 − c)v 1z , the eigenvalue problem is singular. This leads to singular eigenmodes with continuous spectrum in addition to well-behaved eigenmodes with discrete spectrum [14,29].
The continuous spectrum locates at c = v 0 and is real with logarithmic divergence for the mode structure. The discrete spectrum could be either real or complex. where P = diag(1, −1, −1, −1) and T = diag (−1, 1, 1, 1). In general, we expect that physics is invariant with respect to SO + (1, 3), but not with respect to P transformation or T transformation. The program initiated by Bender is to investigate the interesting physics associated with PT transformation [30]. It was demonstrated that in classical systems governed by Newton's second law, such as the dynamical systems in neutral fluids and plasmas, PT symmetry is a consequence of reversibility [13,31]. When a system is not subject to any dissipation, the dynamics is reversible and admits PT symmetry. Note that this observation is consistent with Bender's characterization of PT symmetry as a mechanism of balanced grain and loss for two coupled subsystems [2, 3,10,21,26,28]. If the loss of one subsystem is balanced with the gain of the other subsystem, then the whole system is free of dissipation.
For the KH instability investigated, if a viscosity term µ∇ 2 v is included in Eq.
(2), the coefficients of the reduced ODE do not satisfy the condition (17). In the M → 0 limit, the reduced ODE is the Orr-Sommerfeld equation with complex coefficients [32]. In these cases, the spectrum is not symmetric with respect to the real axis. From our analysis above, it is clear now that the physics here is that viscosity renders the system irreversible and explicitly breaks PT symmetry.
In addition to the properties of spectrum, PT-symmetry analysis also leads to more detailed information about the instability previously unknown. The condition of unbroken symmetry is for some λ. Therefore,f where f = v 1x /p 1 and g = v 1z /p 1 . Equation (28) requires that when the system has unbroken PT symmetry, f is real and g is imaginary. It means that if the system is stable, v 1z should always have a π/2 phase difference relative to v 1x and p 1 . When the system is unstable, PT symmetry is spontaneously broken and PT ψ = λψ does not hold. Thus, the phase differences between these components become arbitrary. Interestingly, similar effects were also observed in optical systems [33], where the PT-symmetric system consists of two coupled waveguides. In these experiments, when PT symmetry was unbroken, the phase difference between two waveguides could be an arbitrary value between [0, π]; when PT symmetry is broken, the phase difference was locked at π/2. proper limits. This algorithm is applicable to both well-behaved modes on the discrete spectrum and singular modes on the continuous spectrum [35]. The numerically calculated stability diagram in the M -k plane is shown in Fig. 1. In the upper (green) region, the system is stable with unbroken PT symmetry, and in the lower (red) region, the system is unstable with spontaneously broken PT symmetry. For this problem, the system is stable with unbroken PT symmetry on the boundary between upper and lower regions.
To explicitly verify PT-symmetry breaking as the mechanism for the KH instability, Therefore, f and g are complex with both real and imaginary parts and vary as functions of z.
In conclusion, together with [13], we have proved that the KH instability is the result of spontaneous PT-symmetry breaking. The discovery of PT symmetry in the KH instability provides a new perspective in the study of classical instabilities in conservative systems. The PT-symmetry analysis for matrix differential operators developed in the present study is applicable to a broader range of systems. We expect that all classical conservative systems are PT-symmetric, and spontaneous PT-symmetry breaking is a generic mechanism for the onset of instabilities in these systems.