Non-Hermitian dynamics of a two-spin system with $\mathcal{PT}$ symmetry

Effective non-Hermitian (NH) Hamiltonians 2018_NatPhys_ElGanainyMakris ; 2021_RevModPhys_Bergholtz have been employed for the description of classical 2016_RMP_KonotopJiankeZezyulin ; 2019_Science_MiriAlu and quantum systems 2007_RepProgPhys_Bender ; 2009_JPA_Rotter that are coupled to the environment by dissipative or other forces. The observation that NH Hamiltonians that are invariant under the combination of parity and time reversal ($\mathcal{PT}$) foster real spectra 1998_PRL_BenderBoettcher ; 2007_RepProgPhys_Bender has led to a series of theoretical and experimental studies. Significant amount of work has focused on optical systems with balanced gain and loss such as in optical beam propagation 2010_NatPhys_RueterMakris and single-mode lasers 2014_Science_FengWongMa . More recently, magnetic systems with $\mathcal{PT}$ symmetry have been proposed 2016_PRB_GaldaVinokur ; 2017_SciRep_GaldaVinokur ; 2019_SciRep_GaldaVinokur ; 2020_PhysD_Barashenkov and systems with gain and loss 2018_PRL_YangWang ; 2021_PRAppl_WangGuoBerakdar ; 2021_arXiv_WittrockPerna ; 2022_arXiv_DengLiFlebus , with the latter reports focusing on degeneracies, called exceptional points (EP), in the spectrum of the linear system, were the sensitivity of the system is enhanced. Effects of EPs were demonstrated for magnonic systems which possess two different loss factors 2019_SciAdv_LiuSun .

Spin systems are studied extensively in relation to magnetic materials HubertSchaefer motivated by applications in magnetic recording, nanoscopic sensors, antennas 2017_NatRevMater_FertReyrenCros , computing applications and neural network implementations 2016_IEEE_GrollierQuerliozStiles . Probing ferromagnets (and other magnetic materials) is done most efficiently by spin torques that are due to a spin-polarized current. This acts on the magnetic moments in the material in a way that may combine energy gain and loss and it drives magnetization dynamics 2008_JMMM_BerkovMiltat . Spin-transfer torque nano-oscillators (STNO) can be constructed that give rise to magnetization oscillators due to injection of dc spin-polarized current RussekRippardCecil_2010 . A $\mathcal{PT}$-symmetric system of this kind with a single spin was studied in Ref. 2016_PRB_GaldaVinokur . The synchronization of chains of STNOs in order to produce large power is a main challenge in this area 2017_NatPhys_AwadAkerman ; 2017_NatComms_LebrunCros .

We propose a $\mathcal{PT}$-symmetric system of two interacting spins or magnetic moments that can realistically be constructed when we invoke suitable spin torque effects. We show that this is described by a complex function instead of a real Hamiltonian. This gives a paradigm of a realistic nontrivial system on the Bloch sphere. We show that the nonlinear system exhibits oscillating motion that is connected with the real eigenvalues of the linearized system. For the full nonlinear system, we derive two conserved quantities that make an explicit connection between the NH system and the Hamiltonian one, and they also furnish a geometric description of the spin trajectories in phase space.

Single spin systems are crucial for quantum computing, and observations on a $\mathcal{PT}$-symmetric single-spin system have been reported 2019_Science_WuLiu . An effective spin Hamiltonian is obtained in various physical systems. An example is a spin Hamiltonian obtained for qubits in an exciton polariton condensate, which are externally controllable by applied laser pulses and are coupled via a coherent tunneling term 2020_NPJ_GhoshLiew . Finally, it is well-known that the possible quantum states for a single qubit can be represented on a Bloch sphere. Our analytical results on complex Hamiltonians could provide tools for a more complete and elegant description of the dynamics of interacting spins on the Bloch sphere in magnetic and other systems.

The paper is organised as follows. In Sec. II, we give the formulation of a spin system using a complex functional. In Sec. III, we give the complex function for a system of two spins coupled by exchange. In Sec. IV, we give analytically simple periodic solutions for the spin dynamics. In Sec. V, we derive two conserved quantities and give their geometric meaning. Sec. VI contains our concluding remarks.

We consider a spin system described via the magnetization vector $\bm{m}=(m_{x},m_{y},m_{z})$ assumed to have a fixed length normalized to unity, $|\bm{m}|=1$. If $E$ denotes the magnetic energy, the conservative torque on the magnetization is $\bm{f}=-\partial E/\partial\bm{m}$. A polarized spin current that is injected in the system may produce an additional torque and the normalised equation of motion is 2008_JMMM_BerkovMiltat

$$\frac{\partial\bm{m}}{\partial t}=\bm{m}\times\frac{\partial E}{\partial\bm{m}}+\alpha\,\bm{m}\times\frac{\partial\bm{m}}{\partial t}-\beta\,\bm{m}\times(\bm{m}\times\bm{p}),$$

(1)

where $\bm{p}$ is the spin current polarization, $\beta$ is a parameter proportional to the polarized current, and we have included a Gilbert damping term with parameter $\alpha$. In the case of a ferromagnetic material, the time variable is scaled to $1/(\gamma\mu_{0}M_{s})$ where $M_{s}$ is the saturation magnetization and $\gamma$ is the gyromagnetic ratio.

The stereographic projection from the unit sphere $|\bm{m}|=1$ to the complex plane is given by the complex variable

$$\Omega=\frac{m_{x}+im_{y}}{1+m_{z}}.$$

(2)

This can be inverted to give the magnetization components

$$m_{x}=\frac{\Omega+\overline{\Omega}}{1+\Omega\overline{\Omega}},\;\;m_{y}=\frac{1}{i}\frac{\Omega-\overline{\Omega}}{1+\Omega\overline{\Omega}},\;\;m_{z}=\frac{1-\Omega\overline{\Omega}}{1+\Omega\overline{\Omega}},$$

(3)

where the bar denotes complex conjugation. The stereographic projection variable will make manifest the non-Hermiticity of the model and it will be central in the formulation of the present work.

The equation of motion for $\Omega$, equivalent to Eq. (1), reads

$$(i+\alpha)\,\dot{\Omega}=-\frac{1}{2}\,(1+\Omega\overline{\Omega})^{2}\,\left(\frac{\partial E}{\partial\overline{\Omega}}+i\beta\,\frac{\partial E_{\rm ST}}{\partial\overline{\Omega}}\right)$$

(4)

where we have defined the function

$$E_{\rm ST}=-\bm{m}\cdot\bm{p}.$$

(5)

Formula (4) suggests the definition of a complex function that includes the spin torque term,

$$F=E+i\beta E_{\rm ST},$$

(6)

so that the equation of motion for $\Omega$ takes the compact form

$$(i+\alpha)\,\dot{\Omega}=-\frac{1}{2}\,(1+\Omega\overline{\Omega})^{2}\,\frac{\partial F}{\partial\overline{\Omega}}.$$

(7)

The function $F$ will be called the complex Hamiltonian.

As a basic example, if we have an external magnetic field $\bm{h}=(0,0,h)$ giving rise to the energy $E=-\bm{m}\cdot\bm{h}$, and the spin polarization is $\bm{p}=(0,0,1)$, then $F=-(h+i\beta)m_{z}$ and the equation of motion is

$$(i+\alpha)\,\dot{\Omega}=-(h+i\beta)\Omega.$$

(8)

We proceed to consider two magnetization vectors or spins interacting via exchange. These may represent two domains or layers in a magnetic material 2020_NatMater_LegrandCrosFert , or two effective spins in another physical system.

We denote the two vectors by $\bm{m}_{1},\bm{m}_{2}$ and assume that they have equal length, $|\bm{m}_{1}|=|\bm{m}_{2}|=1$. Except for the exchange interaction, we include an easy-axis anisotropy along $z$, and an external field $\bm{h}=h\bm{\hat{e}}_{z}$. The energy is

$$E=-J\,\bm{m}_{1}\cdot\bm{m}_{2}-\frac{\kappa}{2}\left[\left(m_{1,z}\right)^{2}+\left(m_{2,z}\right)^{2}\right]-h\,(m_{1,z}+m_{2,z})$$

(9)

where $J>0,\,\kappa>0$ are the exchange and anisotropy parameters respectively, and $m_{1,z},m_{2,z}$ denote the $z$ components of the two vectors. We further consider that spin-polarized currents are injected in the two domains $\bm{m}_{1},\bm{m}_{2}$, with polarization in two opposite directions, $\bm{p}=\pm\bm{\hat{e}}_{z}$. (Such a strategy for achieving $\mathcal{PT}$ symmetry is mentioned in 2016_RMP_KonotopJiankeZezyulin and attributted to 2016_Gaididei .) The opposite polarizations could be achieved, e.g., in the case of two coupled ferromagnetic layers, by injecting a current through the layer separating the two magnetic layers 2021_PRAppl_WangGuoBerakdar , or by two different currents through layers adjacent to each one of the magnetic layers. Spin current polarization with a component perpendicular to the sample plane was recently demonstrated 2022_NatElectr_BoseSchreiberRalph ; 2022_NatElectr_BoseRalph . The equations of motion are

$$\begin{split}\dot{\bm{m}}_{1}&=-\bm{m}_{1}\times\bm{f}_{1}+\alpha\bm{m}_{1}\times\dot{\bm{m}}_{1}-\beta\bm{m}_{1}\times(\bm{m}_{1}\times\bm{\hat{e}}_{z})\\ \dot{\bm{m}}_{2}&=-\bm{m}_{2}\times\bm{f}_{2}+\alpha\bm{m}_{2}\times\dot{\bm{m}}_{2}+\beta\bm{m}_{2}\times(\bm{m}_{2}\times\bm{\hat{e}}_{z})\end{split}$$

(10)

where the dot denotes time differentiation and the effective fields are

$$\begin{split}\bm{f}_{1}&=-\frac{\partial E}{\partial\bm{m}_{1}}=J\bm{m}_{2}+\kappa\,m_{1,z}\bm{\hat{e}}_{z}+h\,\bm{\hat{e}}_{z},\\ \bm{f}_{2}&=-\frac{\partial E}{\partial\bm{m}_{2}}=J\bm{m}_{1}+\kappa\,m_{2,z}\bm{\hat{e}}_{z}+h\,\bm{\hat{e}}_{z}.\end{split}$$

We will assume in the following $h>0,\,\beta>0$. Eqs. (10) have four fixed points. Two fixed points correspond to both $\bm{m}_{1},\bm{m}_{2}$ pointing at the north or at the south pole,

		$\displaystyle\bm{m}_{1}=\bm{m}_{2}=\bm{\hat{e}}_{z}$		(P1)
		$\displaystyle\bm{m}_{1}=\bm{m}_{2}=-\bm{\hat{e}}_{z}$		(P2)

and two further fixed points correspond to the vectors pointing at opposite poles,

		$\displaystyle\bm{m}_{1}=\bm{\hat{e}}_{z},\quad\;\;\,\bm{m}_{2}=-\bm{\hat{e}}_{z}$		(P3)
		$\displaystyle\bm{m}_{1}=-\bm{\hat{e}}_{z},\quad\bm{m}_{2}=\bm{\hat{e}}_{z}.$		(P4)

For the study of the stability of the fixed points as well as for using the concepts of non-Hermiticity, we proceed to the formulation of the system using the stereographic projections (2) of each of the spins, denoted by $\Omega_{1},\Omega_{2}$. We write the complex Hamiltonian of Eq. (6) for the system of two spins,

$$\begin{split}F=&-J\,\bm{m}_{1}\cdot\bm{m}_{2}-\frac{\kappa}{2}\left[\left(m_{1,z}\right)^{2}+\left(m_{2,z}\right)^{2}\right]\\ &-h\,(m_{1,z}+m_{2,z})+i\beta\,(m_{2,z}-m_{1,z}).\end{split}$$

(11)

This can be written in terms of the stereographic variables using Eqs. (3). For example, the exchange term is

$$-J\bm{m}_{1}\cdot\bm{m}_{2}=2J\frac{(\Omega_{1}-\Omega_{2})(\overline{\Omega}_{1}-\overline{\Omega}_{2})}{(1+\Omega_{1}\overline{\Omega}_{1})(1+\Omega_{2}\overline{\Omega}_{2})}.$$

(12)

The equations of motion for $\Omega_{1},\Omega_{2}$, from Eq. (7), read

$$\begin{split}(i+\alpha)\,\dot{\Omega}_{1}=J\frac{1+\Omega_{1}\overline{\Omega}_{2}}{1+\Omega_{2}\overline{\Omega}_{2}}(\Omega_{2}-\Omega_{1})&-\kappa\frac{1-\Omega_{1}\overline{\Omega}_{1}}{1+\Omega_{1}\overline{\Omega}_{1}}\,\Omega_{1}\\ &-(h+i\beta)\Omega_{1}\\ (i+\alpha)\,\dot{\Omega}_{2}=J\frac{1+\overline{\Omega}_{1}\Omega_{2}}{1+\Omega_{1}\overline{\Omega}_{1}}(\Omega_{1}-\Omega_{2})&-\kappa\frac{1-\Omega_{2}\overline{\Omega}_{2}}{1+\Omega_{2}\overline{\Omega}_{2}}\,\Omega_{2}\\ &-(h-i\beta)\Omega_{2}.\end{split}$$

(13)

System (13) is $\mathcal{PT}$-symmetric when we neglect the damping term by setting $\alpha=0$. The terms with $\beta$ act as gain and loss, but note that not each separate one of them can be classified as giving only gain or only loss. This is unlike in previous works on magnetics 2015_PRB_LeeKottosShapiro ; 2018_PRL_YangWang ; 2021_PRAppl_WangGuoBerakdar ; 2022_arXiv_DengLiFlebus , where damping and anti-damping torques were assumed 2022_arXiv_HurstFlebus .

The solution of Eq. (13) $\Omega_{1}=0,\,\Omega_{2}=0$ corresponds to the fixed point (P1) where both vectors point at the north pole. We can study the behavior of the system close to the fixed point if we assume $|\Omega_{1}|,|\Omega_{2}|\ll 1$ and linearize the equations. We have the linearized system (for $\alpha=0$)

$$\begin{split}i\dot{\Omega}_{1}&=-(\omega_{0}+i\beta)\Omega_{1}+J\Omega_{2},\qquad\omega_{0}=J+\kappa+h\\ i\dot{\Omega}_{2}&=-(\omega_{0}-i\beta)\Omega_{2}+J\Omega_{1}\end{split}$$

(14)

that has the form of a standard linear $\mathcal{PT}$-symmetric dimer model. For $\beta\leq J$, we define an angle $\theta$ via

$$\sin\theta=\frac{\beta}{J},\qquad 0\leq\theta\leq\pi$$

(15)

and have a solution

$$\Omega_{1}=A\,e^{i\omega t},\quad\Omega_{2}=A\,e^{i\theta}\,e^{i\omega t},$$

(16)

with angular frequency

$$\omega=\omega_{0}-J\cos\theta.$$

(17)

This solution corresponds to a periodic motion of both spins with equal amplitudes $|\Omega_{1}|=|\Omega_{2}|$. Equivalently, one can say that the spins have equal $z$ components, $m_{1,z}=m_{2,z}$ while precession occurs around the $z$ axis.

For $\beta>J$, the angular frequency has an imaginary part,

$$\omega=\omega_{0}\pm iJ\sqrt{(\beta/J)^{2}-1},$$

(18)

which means that $|\Omega_{1}|,|\Omega_{2}|$ will generically grow in time and, thus, the fixed point is unstable. A corresponding analysis for the fixed point (P2) gives the same stability results, that is, precessional motion for $\beta\leq J$ and an instability for $\beta>J$.

In order to study the fixed point (P3), we use the transformation $\Psi_{2}=1/\overline{\Omega}_{2}$ for the second spin. Then, (P3) corresponds to $\Omega_{1}=0,\Psi_{2}=0$. The equations of motion (13) become (for $\alpha=0$)

$$\begin{split}i\,\dot{\Omega}_{1}=J\frac{1-\Omega_{1}\overline{\Psi}_{2}}{1+\Psi_{2}\overline{\Psi}_{2}}(\Omega_{1}+\Psi_{2})&-\kappa\frac{1-\Omega_{1}\overline{\Omega}_{1}}{1+\Omega_{1}\overline{\Omega}_{1}}\,\Omega_{1}-(h+i\beta)\Omega_{1}\\ i\,\dot{\Psi}_{2}=J\frac{\overline{\Omega}_{1}\Psi_{2}-1}{1+\Omega_{1}\overline{\Omega}_{1}}(\Omega_{1}+\Psi_{2})&-\kappa\frac{\Psi_{2}\overline{\Psi}_{2}-1}{\Psi_{2}\overline{\Psi}_{2}+1}\,\Psi_{2}-(h+i\beta)\Psi_{2}.\end{split}$$

(19)

In order to study stability, we assume $\Omega_{1},\Psi_{2}\ll 1$ and linearize the equations to obtain

$$\begin{split}i\dot{\Omega}_{1}&=(J-\kappa-h-i\beta)\Omega_{1}+J\Psi_{2}\\ i\dot{\Psi}_{2}&=(-J+\kappa-h-i\beta)\Psi_{2}-J\Omega_{1}.\end{split}$$

(20)

Assuming solutions

$$\Omega_{1}=A_{1}e^{i\omega t},\quad\Psi_{2}=\frac{1}{\overline{\Omega}_{2}}=A_{2}e^{i\omega t},$$

we obtain the condition

$$(\omega-h-i\beta)^{2}=\kappa(\kappa-2J).$$

(21)

In the case

$$\kappa\geq 2J\Rightarrow\omega=h\pm\sqrt{\kappa(\kappa-2J)}+i\beta,$$

(22)

we have that $i\omega$ has a negative real part, $-\beta$, giving asymptotic stability for every $\beta>0$. In the case

$$\kappa<2J\Rightarrow\omega=h+i\left[\beta\pm\sqrt{\kappa(2J-\kappa)}\right],$$

(23)

we have that

$$\beta\geq\sqrt{\kappa(2J-\kappa)}$$

(24)

gives asymptotic stability, while

$$\beta<\sqrt{\kappa(2J-\kappa)}$$

(25)

gives instability. The right side in Eq. (23) has a maximum equal to $J$ at $\kappa=J$. This means that, for $\beta>J$, the fixed point is stable for every value of $\kappa$.

Finally, for the fixed point (P4), we can follow a similar procedure and find that it is unstable for every value of $\beta>0,\,\kappa>0$. Table 1 summarizes the results for the stability of the fixed points.

The linear stability results of this section determine cases where fixed points are unstable. On the other hand, in the cases where the linear system is stable (with a real frequency), no conclusive result can be drawn for the nonlinear system (as dictated by standard dynamical systems theory). Further study of the stability will be given in Sec. IV.2.

A system of interacting spins that are under spin transfer torque can be described by a complex function, or a non-Hermitian Hamiltonian. We give the formalism for obtaining the complex function that makes manifest the $\mathcal{PT}$ symmetry when this is present. We have studied the nonlinear dynamics dynamics and the exceptional point for a system of two exchange-coupled spins in a case of $\mathcal{PT}$ symmetry. This introduces a paradigm for the dynamics of non-Hermitian systems defined on the sphere.

We have identified the regime for periodic precessional motion of the two spins, and its stability. The periodic motion corresponds to a locking of the complete system in sustained magnetization oscillations. The synchronization of the oscillations of the two spins, due to $\mathcal{PT}$ symmetry, could lead to an answer to the long standing problem of the synchronization of spin-transfer torque nano-oscillators (STNO) 2017_NatPhys_AwadAkerman ; 2017_NatComms_LebrunCros . The rich spin dynamics for the spin system can be further used to study non-magnetic systems which are described by effective spin variables (e.g., as in polariton condensates 2020_NPJ_GhoshLiew ).

The author is grateful to Kostas Makris for discussions and insightful remarks. This work was supported by the project “ThunderSKY” funded by the Hellenic Foundation for Research and Innovation and the General Secretariat for Research and Innovation, under Grant No. 871.