Abstract
Magnetic systems governed by exchange interactions between magnetic moments harbor frustration that leads to ground state degeneracy and results in the new topological state often referred to as a frustrated state of matter (FSM). The frustration in the commonly discussed magnetic systems has a spatial origin. Here we demonstrate that an array of nanomagnets coupled by the real retarded exchange interactions develops a new state of matter, time frustrated matter (TFM). In a spin system with the timedependent retarded exchange interaction, a single spinflip influences other spins not instantly but after some delay. This implies that the sign of the exchange interaction changes, leading to either ferro or antiferromagnetic interaction, depends on time. As a result, the system’s temporal evolution is essentially nonMarkovian. The emerging competition between different magnetic orders leads to a new kind of timecore frustration. To establish this paradigmatic shift, we focus on the exemplary system, a granular multiferroic, where the exchange transferring medium has a pronounced frequency dispersion and hence develops the TFM.
Introduction
Macroscopic magnetism forms due to microscopic exchange interactions^{1,2,3}. The exchange interaction is of the quantum mechanical origin and stems from the intertwined effect of the Coulomb interaction and Pauli exclusion principle governing the behavior of indistinguishable fermions with overlapping wave functions. Remarkably, a rich lore narrating how the geometric frustration developing from the exchange effects leads to degeneracies and the emergent FSM, neglects the temporal component: local magnetic moments are supposed to instantly interact with each other without delays^{4}. However, the flip of electron spin in an atom in a crystal implies a rearrangement of electron density distribution in space, which, in turn, affects the strength of interactions between the atom with its neighbors. The rearrangement of electron clouds occurs at optical frequencies (\( \sim 500~\mathrm {THz}\)) and so with characteristic time scales \(\sim 0.01~\mathrm {ps}  0.1~\mathrm {ps}\), while the relaxation of the atomic position caused by the spin flip occurs at phonon frequencies (\(\sim 1~\mathrm {THz}\)) and so picosecond time scales^{5}. As a result, the spin exchange interaction in solids should be in general nonlocal in time and has the time delayed (retarded) nature.
There is an abundance of new functional materials, like granular multiferroics^{6,7}, where interactions occur not directly but through the mediating active dielectric or ferroelectric environment. In such materials^{8,9,10,11} the retardation effects are relatively large and cannot be ignored. There, the polarization, \({\mathbf {P}}\), of a ferroelectric manifests retarded response to the electric filed \({\mathbf {E}}\), hence \(\mathbf{P}(t)=\int {\hat{\alpha }}(tt'){\mathbf {E}}(t')dt'\), where \({\hat{\alpha }}\) is the polarizability tensor^{12,13} in a linear response approximation. The superexchange interaction of magnetic moments in granular multiferroics^{14}, where electric and magnetic degrees of freedom mutually influence each other, acquires retardation as well. In our work we reveal the time retardation of the exchange interaction and investigate the timefrustrated state of matter emerging due to this retardation in an array of magnetic moments immersed into the ferrolectric environment.
Relaxation of the exchange has recently been intensely studied, both experimentally and theoretically^{15,16,17,18,19,20,21,22,23}. One of the most discussed examples has been relaxation of the system after an instantaneous external action of, for example, laser irradiation finiteduration pulse. Here, we address a different situation where the retardation emerges due to internal properties of the system. The resulting relaxation manifests a variety of the nontrivial effects appearing even without the external finitetime impacts. We are confident that our finding on the retarded nature of exchange would contribute to investigations of relaxation processes in the timedependent exchange.
The model
To reveal how the exchange retardation results in the TFM we focus first on an elemental building block of a granularmultiferroic, two adjacent magnetic granules interacting via a ferroelectric medium as schematically shown in Fig. 1. Figure 1a displaces two metallic granules carrying the opposite magnetic moments disposed over the ferroelectric substrate and Fig. 1b presents the same magnetic moments immersed into a ferroelectric medium. Since the dielectric constant of a ferroelectric environment typically has significant frequency dispersion, the retardation effects are inevitable. Indeed, in a simplest approximation taking into account the dielectric screening of the Coulomb interaction, one finds, following^{10,11,24}, that the dielectric constant appears in the effective exchange between two magnetic moments as
where the sum is taken over the electron wave functions of each granule, and \(\Psi _{a,b} ({\mathbf {r}}_{1,2} )\) stand for the undisturbed electron wave functions. More precise calculation requires including the effect of the environment on the wave functions^{24} and also accounting for the spacial dispersion effects. Yet, even in this first approximation, the frequency dispersion of the exchange integral \(J(\omega )\) arises due to dispersion of \(\varepsilon (\omega )\) the behavior of which is straightforwardly related to the dielectric permittivity tensor of the ferroelectric environment^{10,11,14,24,25}. Accordingly, we arrive at the model of a magnetic system with the effectively delayed exchange. In a temporal representation, this implies the delay in the interaction of magnetic moments: \(\int J_{12}(tt')\,{{\mathbf {m}}}_{1}(t)\,{{\mathbf {m}}}_{2}(t')dt'\), where \(J_{12}(tt')\) is the Fourier transform of \(J(\omega )\). It is essential that \(J_{12}(tt')\) is purely retarded, that is \(J_{12}(tt') =0\), if \(t<t'\), so the causality is fulfilled.
In typical ferroelectrics (e.g., such as barium titanate (BTO) and lead zirconate titanate (PZT)), \(\varepsilon (\omega )\) is large at low frequencies, \(\varepsilon \,(\omega =0)= \varepsilon _0\gtrsim 1000\), and is of the order of unity for large frequencies, \(\varepsilon \,(\omega =\infty )= \varepsilon _{\infty }\simeq 1\). The frequency threshold is set by the phonon frequency which usually does not exceed 1 THz. Consequently, for small frequencies, we may treat \(J(\omega )\) as vanishing, while at large frequencies, \(J(\omega )\) tends to finite values.
Accordingly, we put
implying that the function J(t) is alternating in sign with time. Hence we arrive at the “time frustration” of the exchange interaction.
Let us consider now two adjacent magnetic moments \({\mathbf {m}}_{1}\), \({\mathbf {m}}_{2}\). The delay in the interaction implies a nonequilibrium regime at finite times, while due to the nonzero damping the magnetic moments assume stationary values, \({\mathbf {m}}_{1,2}^{(\infty )}\), at \(t\rightarrow \infty \). The final magnetic state is to be derived from the energy considerations using the effective exchange Hamiltonian \(H=J_{12}(\omega =0)\, {\mathbf {m}}_{1}^{(\infty )} {\mathbf {m}}_{2}^{(\infty )}\). The mutual orientation of magnetic moments is to be found by investigating the magnetization time evolution from the starting point to \(t\rightarrow \infty \).
The magnetic granules are supposed to be semiclassical, hence granule’s magnetization should obey the nonlocal in time LandauLifshitzGilbert (LLG) equation. We consider an array of localized magnetic moments satisfying \({\mathbf {m}}_i(t)=1\) condition. The equation of motion for ith moment is
were, as usual, \(\gamma \) is the gyromagnetic ratio and \(\lambda \) is the damping parameter. Here \({\mathbf {h}}_i^{\mathrm{{eff}}}\) is an effective Weiss field at the ith cite defined as
where \({\mathbf {h}}^{\mathrm{{ext}}}(t)\) is the weak external magnetic field, and the sum runs over the nearest neighbors to the site \({\mathbf {i}}\) (we account for only the nearestneighbor exchange interactions). Exchange integrals \(J_{\mathbf {ij}}(t)\) are timedependent and preserve the causality. An instant interaction \(J_{\mathbf {ij}}(tt')=\delta (tt'0)J_{\mathbf {ij}}\) corresponds to the standard LLG equation^{26,27} implying that the exchange energy assumes the usual form, \(1/2 \sum _{{\mathbf {i}},{\mathbf {j}}} J_{\mathbf {ij}}\ \mathbf {m_i}\, \mathbf {m_j}\).
Stationary solutions to the LLG equation
Let us consider a stationary solution to Eq. (3) in which we hereafter set \(\gamma = 1\) for simplicity. We assume that \({\mathbf {m}}_{i}(t)={\mathbf {m}}^{0}_{i}\) is a set of stationary solutions to Eq. (3) in the absence of the external field, \({\mathbf {h}}^{\mathrm{{ext}}}(t) = 0\). For better visibility we further simplify the notations and write:
where \(\sum _{NN}\) stands for the sum over the nearest neighbors. In this transparent case, the LLG equation assumes the form
which requires that \({\mathbf {b}}_{i}^{0} = 0\). Stationary solutions would realize for either

Frustrated exchange with \(J_{0}=\int _{\infty }^{t \rightarrow \infty }J(t\tau )d\tau =0\) implying \({\mathbf {h}}_{i}^{0}\,\mathbf {\equiv 0}\), hence \({\mathbf {b}}_{i}^{0}\,\mathbf {\equiv 0}\). Therefore, any magnetic configuration formally assumes a stationary solution. We address further the important question whether these solutions are stable with respect to small perturbations like noise or an external field, and show that only particular stationary configurations are stable.
or for

A nonfrustrated exchange with \(J_{0}=\int _{\infty }^{t \rightarrow \infty }J(t\tau )\,d\tau \ne 0\), where we have a condition \({\mathbf {b}}_{i}^{0}\,= J_{0} [{{\mathbf {m}}_{i}}\times {\sum _{NN} \mathbf {m_{NN}}}] = 0\) , which is satisfied for common FM, AFM structures and for collinear configurations for which \(\sum _{NN} \mathbf {m_{NN}}=0\), e.g., stripe structures in square lattice.
We see that for stationary solutions, frustrated and nonfrustrated cases differ qualitatively. Hereafter we focus on a dynamically frustrated case.
Timedependent magnetic moments
General equations
Let us derive the adjustments to the stationary solution discussed above arising due to time dependence of the magnetic moments. We consider the timedependent part of the ith magnetic moment \({\mathbf {m}}^{\delta }_{i}(t)\) being small and write magnetization as
In the corresponding linear approximation, the LandauLifshitzGilbert equation for \({\mathbf {m}}^{\delta }_{i}(t)\) in the frustrated case assumes the form, see Methods
To find the analytical solution to Eq. (10) we take its Fourier transform and obtain, see Methods,
Twosite cluster with the frustrated exchange
The above general reasoning holds for any arbitrary regular magnetic structure and, in particular, is not restricted to systems subject to nearest neighbor interactions constraint. To illustrate how the formation of the timefrustrated state occurs, we consider the simplest particular system, two interacting magnetic moments \({\mathbf {m}}_{1}\) and \({\mathbf {m}}_{2}\). To further simplify the problem, we analyze collinear stationary configurations, FM with \({{\mathbf {m}}_{1}^{0} = {\mathbf {m}}_{2}^{0}\parallel z}\) and AFM with \({{\mathbf {m}}_{1}^{0} =  {\mathbf {m}}_{2}^{0}\parallel z}\) (hereafter we use \({\mathbf {m}}_{i}^{0}\cdot {\mathbf {m}}_{i}^{0}=1,\ i=1,2\) and \({\mathbf {m}}_{1}^{0}\times {\mathbf {m}}_{2}^{0} = 0\)).
Taking the simplest form of the exchange satisfying all the abovedefined conditions
where \(\delta (t)\) is the Dirac delta function using its Fourier transform,
see Fig. 2, which preserves the causality as the pole is in the lower half of the \(\omega \)plane, and displays a reasonable asymptotic behavior: \(J(\omega =0)=0\), \(J(\omega \rightarrow \infty )\rightarrow \text {const}\).
To simplify further notations, we set it in that the magnetic moments and energy are properly normalized and are measured in dimensionless units. Thus, G, \(\gamma \) and \(\lambda \) also become dimensionless. Now we find the stability conditions ensuring the stationary solutions. For the FM case, \(m_{1}^{0}=m_{2}^{0}=+1\), we obtain the stability condition as \(G\lambda <\omega _0 \). For the AFM configuration, \(m_{1}^{0} =  m_{2}^{0} = +1\), and the resulting stability condition is \({G}\sqrt{1+\lambda ^{2}} < \omega _0 \).
Having established the ranges of stability within the linear approximation, let us turn to detailed investigating the time evolution of our system. The zerofrequency limit was discussed above. At high frequencies the ferroelectric degrees of freedom are frozen and the exchange is provided by the conventional electron clouds overlapping.
The time evolution appears radically different in the isotropic case and in the presence of the even weak uniaxial anisotropy. In the isotropic case, the asymptotic, \(t \rightarrow \infty \) state of magnetic moments, either in the FM or AFM case, is defined by the exchange potential parameters, mostly by its \(\delta \)part. In the anisotropic case either the \(\delta \)part or the exponential part of the timedepending exchange (12) dominates the system’s behavior. The results of the numerical calculations of the time evolution are displayed in Fig. 3. The panel Fig. 3a shows that the evolution of the isotropic system with the timedependent exchange results in the FM state at \(t\rightarrow \infty \), as it is clearly seen from the evolution of magnetic moments projections. Remarkably, although perturbing the system by the halfsinusoidal pulse of the external magnetic field switches the FM state into the AFM one, see the panel Fig. 3b, this AFM state lives only for some finite time, and then the system returns back to the FM state.
This whole evolution picture is noiseresistant, it does not transform under the deltacorrelated noise with the amplitude \(\mathbf {h_{noise}}\) small relative to effective field \({\mathbf {h}}_{noise} \ll {\mathbf {h}}^{\mathrm{{ext}}}(t)\). The external perturbation in the form of the sequence of the alternating pulses successively converts FM to AFM and vice versa, see Supplementary Information (SI).
In the presence of the anisotropy, both states, the AFM and the FM, become stable. There exist two ways of switching the final destination of the system between these states. The first way is changing the parameters of the dynamically frustrated potential. The example of such a switch by changing the characteristic frequency \(\omega _0\) is shown in Fig. 4a presenting the temporal evolution of quantity \(\mathbf {{m_1}\cdot {m_1}}\), characterizing the state of the system. The second possibility of the switching is the perturbation in a form of the single halfsinusoidal external magnetic field pulse. In this case, in contrast to the isotropic one, the switched state is stable. Depending on the direction of the pulse, the final stable state is either the FM or the AFM state. Figure 4b shows an example of such a switch.
The revealed behaviours are of a general character and maintain for a general case of the system with the arbitrary number of the magnetic moments. The behaviour of the exemplary foursite cluster is presented in the Supplementary Information (SI).
Discussion and conclusion
We have studied spin system with retarded spinspin interaction \(J_{ij}\). This implies the nonMarkovian type of the timedependent magnetic interaction and leads to nontrivial dynamics of the interacting magnetic moments. The timefrustrated case where \(J_{ij}(\omega =0)=\int _0^\infty J_{ij}(t)dt=0\) is the most interesting regime because in this case the sign of \(J_{ij}(\omega =0)\) does not naively predict the arising at \(t\rightarrow \infty \) magnetic configuration.
It is important to stress that the retardation causes the nonHermiticity of the effective Hamiltonian of the interacting magnetic moments, therefore, the considered system is an effectively dissipative. NonHermitian quantum mechanics describing open dissipative systems is currently enjoying an intense explosive development^{28,29,30,31}, and further aspects and implications of the nonHermitian behavior of the system in hand will be a subject of the forthcoming publication.
The retarded spinspin interaction is realized in the systems with the superexchange where magnetic moments interact indirectly through a medium with the pronounced frequency dispersion, granular multiferroics offering an appealing example. In multiferroics, magnetic granules interact through a ferroelectric medium. Its polarization comprises several contributions with the different characteristic times, \({\mathbf {P}}={\mathbf {P}}_{\mathrm{el}}+{\mathbf {P}}_\mathrm{ion}+{\mathbf {P}}_{\mathrm{dipols}} + \ldots \). Here the first “elastic” contribution is the polarization of the outer electron shells, the second one is related to the ion shifts, and the third contribution is related to dipole moments of molecules; the second and the third terms typically are responsible for the ferroelectricity. It is important that all the contributions except the first one are relatively slow, with their relaxation times being larger or of order of the inverse phonon frequencies for which 1 THz is a natural scale^{32,33,34}. At the same time, \(P_{\mathrm{el}}\) relaxation time is electronic, having the optical frequencies, being thus by several orders of magnitude shorter (G in Eq. (12) is obviously define by \(P_\mathrm{el}\)). When magnetic moments evolve fast, the superexchange interaction involves only polarization of the outer electron shells, while slow evolution of magnetic moments involves the change of the ferroelectric polarization due to shift of ions. This is the picture in the frequency domain. In the time domain this physical mechanism leads to the retarded spinspin interaction.
There are other frequency dependent exchange channels in the problem. We consider one of the most important cases. Other known mechanisms either look similar to the considered one or are suppressed by the Coulomb blockade^{35,36}. Note here one must talk about the direct exchange modulated by the environment. We have avoided using this term, preferring a somewhat loose use of the word ”superexchange”. Moreover, in the exact sense of the word, there is no ”complete” direct exchange in our case. At low frequencies, there is no overlap, it occurs only when the excitation of the medium is taken into account.
Let us also make a supporting remark. We have demonstrated the discussed effect for a particular set of parameters. A similar behavior of the system is observed while varying them in the wider range, in particular, when varying \(\lambda \) by an order of magnitude up and down.
Mutliferroics, the systems with interacting magnetic and electric degrees of freedom, broaden the scope of the existing current hardware concepts^{7,37,38,39,40,41,42,43,44,45,46,47,48,49} and introduce the new ones, see^{50,51,52,53}. Magnetic mutliferroic tunnel junctions promise the platform for the computers based on the nonbinary (manyvalued) logics^{54}. We demonstrated that owing to the dynamic frustration, the ferromagnetic and antiferromagnetic are stable and longliving states, hence having, in fact, the same energies. In particular, the system that we have considered renders the tunnel junction magnetferroelectricmagnet with the time frustrated exchange having four different stable states, comprising two ferroelectric and two magnetic states. Hence the computer element based on the TFM holds high potential for the fourvalued logic hardware realizations.
Methods
Derivation of a general equation
Taking magnetization as
where \({\mathbf {m}}^{\delta }_{i}(t)\) is assumed small, one finds that the corresponding linear approximation of the LandauLifshitzGilbert equation for \({\mathbf {m}}^{\delta }_{i}(t)\) assumes the form
In the frustrated case with \(\int _{\infty }^{t \rightarrow \infty } J(t\tau ) d\tau = 0\), both \({\mathbf {h}}_{i}^{0}(t)=0\) and \({\mathbf {b}}_{i}^{0}(t)=0\) for large enough values of t, and Eq. (15) reduces to Eq. (10) of the main text. Taking the Fourier transform of (10), one gets
where
The double cross product in (18) is (\({\mathbf {m}}_{i}^{0} \cdot {\mathbf {m}}_{i}^{0} = 1\))
and the evolution equation for ith for the Fourier transform of magnetic moment becomes Eq. (11) of the main text.
Twosite cluster with the frustrated exchange
For the twosite cluster the system of equations (11) for \({\mathbf {m}}_{1}^{\delta }(\omega )\), \({\mathbf {m}}_{2}^{\delta }(\omega )\) reads (we set \({\mathbf {h}}^{\mathrm {ext}}(\omega ) \parallel x\)):
with (after projection to x and yaxes) the determinant
were \(m_{i}^{0} = \left {\mathbf {m}}_{i}^{0}\right \).
Taking the exchange in the form Eq. (13) that not only preserves the causality, but also \(\mathrm { Re\,} J(\omega )\) is even function of \(\omega \) like \(\mathrm { Re\,}1/\epsilon (\omega )\). Then the characteristic equation \(\Delta (\omega ) = 0\) has four roots (apart from four stationary roots \(\omega _{0} = 0\)):
For the FM case \(m_{1}^{0}=m_{2}^{0}=+1\), we find
so the stability condition is \(G\lambda < \omega _0 \).
For the AFM configuration \(m_{1}^{0} =  m_{2}^{0} = +1\), we have two doubledegenerate roots
and the resulting stability condition is \({G}\sqrt{1+\lambda ^{2}} < \omega _0 \).
Data availability
No special data are generated in this work. Topical subheadings are allowed.
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Acknowledgements
The work of V.M.V was supported by Terra Quantum AG. The work of N.M.Ch. was supported by the Russian Science Foundation (grant 181200438). Numerical calculations were performed using computing resources of the federal collective usage center Complex for Simulation and Data Processing for Megascience Facilities at NRC “Kurchatov Institute” (http://ckp.nrcki.ru/) and supercomputers at Joint Supercomputer Center of RAS (JSCC RAS).
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N.M.C and A.V.M. conceived the work and performed calculations, V.E.V. carried out numerical simulations, V.M.V. took part in calculations and outlining the project, N.M.C, A.V.M., and V.M.V. lead the interpretation of the results and wrote the manuscript, all authors discussed the manuscript.
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Valiulin, V.E., Chtchelkatchev, N.M., Mikheyenkov, A.V. et al. Timedependent exchange creates the timefrustrated state of matter. Sci Rep 12, 16177 (2022). https://doi.org/10.1038/s4159802219751y
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DOI: https://doi.org/10.1038/s4159802219751y
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