Nonreciprocal Frustration: Time Crystalline Order-by-Disorder Phenomenon and a Spin-Glass-like State

Active systems are composed of constituents with interactions that are generically nonreciprocal in nature. Such nonreciprocity often gives rise to situations where conflicting objectives exist, such as in the case of a predator pursuing its prey, while the prey attempts to evade capture. This situation is somewhat reminiscent of those encountered in geometrically frustrated systems where conflicting objectives also exist, which result in the absence of configurations that simultaneously minimize all interaction energies. In the latter, a rich variety of exotic phenomena are known to arise due to the presence of accidental degeneracy of ground states. Here, we establish a direct analogy between these two classes of systems. The analogy is based on the observation that nonreciprocally interacting systems with antisymmetric coupling and geometrically frustrated systems have in common that they both exhibit marginal orbits, which can be regarded as a dynamical system counterpart of accidentally degenerate ground states. The former is shown by proving a Liouville-type theorem. These “accidental degeneracies” of orbits are shown to often get “lifted” by stochastic noise or weak random disorder due to the emergent “entropic force” to give rise to a noise-induced spontaneous symmetry breaking, in a similar manner to the order-by-disorder phenomena known to occur in geometrically frustrated systems. Furthermore, we report numerical evidence of a nonreciprocity-induced spin-glass-like state that exhibits a short-ranged spatial correlation (with stretched exponential decay) and an algebraic temporal correlation associated with the aging effect. Our work establishes an unexpected connection between the physics of complex magnetic materials and nonreciprocal matter, offering a fresh and valuable perspective for comprehending the latter.

Popular Summary

A nonreciprocal interaction is an interaction where action-reaction symmetry is broken. Although such interactions are forbidden in a closed equilibrium system, they can arise once the system is driven out of equilibrium. Here, we establish an unexpected link between this category of nonequilibrium systems to a seemingly unrelated concept: geometrical frustration. We show that exotic phenomena known to occur in the latter also occur in the former.

To see this connection intuitively, imagine two objects that interact with each other. If their interactions are reciprocal, say attractive, they will both be “happy” when they get as close to each other as possible. However, when nonreciprocally interacting, where one object feels an attractive force from another but the other is repulsed by the first, there are no configurations that satisfy both simultaneously. This situation is analogous to those encountered in geometrically frustrated systems, where conflicting objectives also exist, resulting in the absence of configurations that simultaneously minimize all interaction energies.

It is well known that geometrically frustrated systems often exhibit accidental ground-state degeneracies, which give rise to various exotic phenomena. We prove that nonreciprocal interactions can give rise to phase-space orbits that remember their initial conditions, a dynamical analog of accidentally degenerate ground states. We show that these orbits give rise to a dynamical counterpart of frustration-induced phenomena such as the so-called order-by-disorder phenomena and a spin glass.

These results offer a new viewpoint for understanding this class of nonequilibrium systems and serve as a design principle for future devices with nonreciprocity.

Figure 1

Geometrical and nonreciprocal frustration and the emergence of “accidental degeneracy” of orbits. (a1)–(c1) Examples of systems with (a1) no frustration, (b1) geometrical frustration, and (c1) antisymmetric nonreciprocal interactions. (a2),(b2) Schematic description of energy $E$ as a function of the spin configuration $\{{\theta }_j\}$. Here, $E_{\mathrm{G}}$ is the ground state energy and we have omitted the degeneracy that trivially arises from the global rotation symmetry, for clarity of the figure. Note that the energy $E$ is not defined for the nonreciprocal case. (a3)–(c3) Orbits. (a4)–(c4) Lyapunov exponents $\lambda$. (a) In frustration-free systems, since fixing the angle of one spin would determine all other spin configurations to minimize the energy of the system, the ground state is unique up to global symmetry. As a result, the system converges into a unique stable fixed point [red point in (a3)], which gives negative Lyapunov exponents $\lambda <0$. (b) Geometrically frustrated systems, on the other hand, often exhibit accidentally degenerate ground states because of the underconstrained degrees of freedom. The presence of the degenerate ground states implies that there is a direction in which a restoring force (torque) is absent. This means that the accidentally degenerate ground states correspond to marginal fixed points in the language of dynamical systems [blue line in (b3)] that have zero Lyapunov exponent(s) $\lambda =0$. (c) In nonreciprocally frustrated systems with perfect nonreciprocity $J_{ij}=-J_{ji}$, the spins start a chase and run away motion that corresponds to marginal orbits with zero Lyapunov exponents $\lambda =0$, arising due to the Liouville-type theorem [Eq. (3)]. These orbits can be regarded as the dynamical counterpart of the ground state accidental degeneracy of geometrically frustrated systems that also have zero Lyapunov exponents $\lambda =0$.

Figure 2

Marginal orbits in perfectly nonreciprocal three-spin system. We set $J_{12}=-J_{21}=3$, $J_{23}=-J_{32}=-1$, and $J_{31}=-J_{13}=2$.

Figure 3

The concept of order-by-disorder phenomena in equilibrium and their generalization to nonreciprocally interacting systems. (a),(b) Energy and free-energy profiles in a geometrically frustrated system (a) without noise (i.e., zero temperature $T=0$) and (b) with noise (i.e., finite temperature $T>0$) in a geometrically frustrated system. (c),(d) Orbits in a geometrically frustrated system without (c) and with (d) thermal noise. At $T=0$, the system exhibits ground state degeneracy, while the presence of thermal noise at $T>0$ lifts this degeneracy through entropic forces. (e),(f) Orbits in a nonreciprocally interacting system without (e) and with (f) noise. Similarly to the geometrically frustrated system counterpart, “entropic force” selects one of the orbits to give rise to OBDP.

Figure 4

Accidental degeneracy of orbits in geometrically and nonreciprocally frustrated all-to-all coupled many-body systems. The thick arrows represent the macroscopic angles ${\varphi }_a$ that are composed of a macroscopic number of spins represented by smaller solid arrows. A geometrically frustrated four-community system illustrated in (a) exhibits an accidental degeneracy parametrized by a relative angle $\alpha$. Similarly, a nonreciprocally frustrated two-community system illustrated in (b) exhibits marginal orbits parametrized by a relative angle $\mathrm{\Delta }\varphi$. These degeneracies are shown to get lifted by introducing disorder to the system.

Figure 5

Time crystalline order-by-disorder phenomena induced by nonreciprocal frustration. Time evolution of (a) the angle difference $\mathrm{\Delta }\varphi ={\varphi }_{\mathrm{A}}-{\varphi }_{\mathrm{B}}$, (b) ${\varphi }_{\mathrm{A}},{\varphi }_{\mathrm{B}}$, and (c) the effective coupling strength $j_{ab}^{\star }(\varphi )$ [Eq. (12)]. In (a), the solid (thin) line represents the dynamics in the presence (absence) of noise. We set $j_{\mathrm{AA}}=j_{\mathrm{BB}}=3$, $j_{\mathrm{AB}}=-j_{\mathrm{BA}}=1$, the noise strength $\sigma =1.5$, and the number of spins $N_{\mathrm{A}}=N_{\mathrm{B}}=2000$. The system “selects” $\mathrm{\Delta }{\varphi }_{*}=\pm \pi /2$ that satisfies Eq. (20) to give rise to the time-dependent phase (chiral phase), all in agreement with our analytical analysis in the main text.

Figure 6

Noise-induced spontaneous parity breaking. The computed noise strength dependence of the angle difference $\mathrm{\Delta }\varphi ={\varphi }_{\mathrm{A}}-{\varphi }_{\mathrm{B}}$ of the order parameter ${\psi }_a=r_a{e}^{i{\varphi }_a}$ in the steady state for $j_{\mathrm{AB}}=0.35\ne -j_{\mathrm{BA}}=0.25$. Here, the solid lines and the shaded area represent the average and the variance of $\mathrm{\Delta }\varphi$, respectively, which were calculated using the data after reaching a steady state (in the time range $100<t<400$) with an initial condition ${\theta }_i^{\mathrm{A}}(t=0)=\pi /4(=-\pi /4)$ and ${\theta }_i^{\mathrm{B}}(t=0)=0$ for blue (orange) plots. The transition from $\mathrm{\Delta }\varphi =0$ (parity-symmetric static phase) to $\mathrm{\Delta }\varphi \ne 0$ (parity-broken chiral phase) as $\sigma$ increases indicates that the noise has induced the spontaneous symmetry breaking, which is a salient feature of OBDP and is opposite from what is usually expected. We set $j_{\mathrm{AA}}=j_{\mathrm{BB}}=1$, and $N_{\mathrm{A}}=N_{\mathrm{B}}=2000$.

Figure 7

Schematic phase diagram. (a),(b) Schematic steady-state phase diagram in the absence of noise $\sigma =0$ (a) and in the presence of noise $\sigma >0$ (b). The intracommunity couplings $j_{\mathrm{AA}}>0$ and $j_{\mathrm{BB}}>0$ are assumed to be large compared to intercommunity couplings $j_{\mathrm{AB}}$ and $j_{\mathrm{BA}}$.

Figure 8

Order-by-disorder phenomena in a nonreciprocally coupled three-community system with random torque. Solid (thin) lines represent the trajectories for different initial conditions in the presence (absence) of random torque, where one can see that certain orbits are selected by the disorder. The angle difference ${\varphi }_a-{\varphi }_b$ dynamics are computed using the Ott-Antonsen ansatz [69, 70] [Eq. (29) in Appendix pp2]. We set the coupling $j_{\mathrm{AA}}=j_{\mathrm{BB}}=j_{\mathrm{CC}}=4$, $j_{\mathrm{AB}}=-j_{\mathrm{BA}}=3$, $j_{\mathrm{BC}}=-j_{\mathrm{CB}}=-1$, $j_{\mathrm{CA}}=-j_{\mathrm{AC}}=2$, the torque distribution width $\mathrm{\Delta }=0.1$.

Figure 9

Nonreciprocal $XY$-spin system on a hypercubic lattice. In this model, $XY$ spins on a hypercubic lattice interact with their next-nearest-neighbor (nearest-neighbor) spins in a reciprocal (nonreciprocal) manner. The spins are divided into two communities, namely the sublattices A and B. Within each sublattice, the spins interact reciprocally via next-nearest-neighbor interactions $j_{\mathrm{AA}}$ and $j_{\mathrm{BB}}$, while the spins on different sublattices interact nonreciprocally through nearest-neighbor interactions $j_{\mathrm{AB}}$ and $j_{\mathrm{BA}}$. The figure illustrates the case of a two-dimensional spatial dimension ($d=2$), but the model can be readily extended to higher spatial dimensions.

Figure 10

One-dimensional spin chain with random asymmetric nearest-neighbor coupling.

Figure 11

Domain-wall annihilation dynamics in reciprocal one-dimensional random spin chain. The reciprocal coupling $J_i^{\mathrm{R}}=J_i^{\mathrm{L}}$ ($J_{ij}=J_{ji}$) case. As shown in the bottom left-hand panel, the ground state configuration of the reciprocally coupled spin chain (where the signs represent the sign of the reciprocal coupling at each bond) exhibits nematic order that is unique up to global rotation, implying the absence of geometrical frustration. (a) Typical trajectory of (nematic) angles ${\phi }_i={\theta }_i(\text{mod}\pi )$. We set $N=2^9=512$, and the initial conditions were taken randomly from a uniform distribution ${\theta }_i=[0,2\pi )$. (b) Spatial correlation function $C_x(x,t)$. (c) Time correlation function $C_t(t_w+t,t_w)$. In (b) and (c), we have averaged over 400 trajectories of random initial conditions and configurations of coupling strengths and have set $N=2^{10}=1024$. The domain-wall annihilation dynamics of this one-dimensional chain give rise to slow relaxation (that shows aging phenomena) toward a long-ranged nematically ordered state.

Figure 12

Nonreciprocal frustration-induced spin-glass-like state in an asymmetric random spin chain. The asymmetric coupling case ($J_{ij}\ne J_{ji}$) with the coupling to the left $J_i^{\mathrm{L}}$ and the right $J_i^{\mathrm{R}}$ sampled independently ($\gamma =0$). (a),(b) Typical trajectory of (nematic) angles ${\phi }_i={\theta }_i(\text{mod}\pi )$ of a one-dimensional random nonreciprocal spin chain. Here, we have set $N=2^9=512$, and the initial condition was taken randomly from a uniform distribution ${\theta }_i=[0,2\pi )$. (b) Line-cut data of the trajectory at site $i=230$, 233, 236 of (a). (c) Spatial correlation function $C_x(x,t)$. (d) Time correlation function $C_t(t_w+t,t_w)$. Note that both axes are plotted on a logarithmic scale. We have averaged over 500 trajectories for ${\sigma }_{J}t\le 6.4\times {10}^3$ in (c) and 400 trajectories for ${\sigma }_{J}t\ge 3.2\times {10}^4$ in (c) and for all plots in (d). We have set $N=2^{10}=1024$. This system exhibits slow dynamics characterized by power-law decay and aging phenomena and a short-ranged spatial correlation (which exhibits stretched exponential decay; see Fig. 13), where the latter property is in stark contrast to the nematically ordered state seen in the reciprocal case (Fig. 11). These properties are reminiscent of a spin glass.

Figure 13

Stretched exponential decay of spatial correlation function in an asymmetric random spin chain. This is identical to Fig. 12 but plotted in a different scale. The dashed line is a fitted curve $C_x(x,t)=e^{-(x/\xi {)}^{\alpha }}$ to the ${\sigma }_{J}t={10}^5$ result, which gives $\alpha \simeq 0.49$.

Figure 14

Asymmetry parameter $\gamma$ dependence of the spatial and time correlation functions. (a)–(c) Spatial correlation function $C_x(x,t)$ and (d)–(f) temporal correlation function $C_t(t+t_w,t_w)$. (a),(d) $\gamma =1$. (b),(e) $\gamma =0.9$. (c),(f) $\gamma =-0.5$. We set $N=500$ in (a) and (d) and $N=1000$ in (b), (c), (e), and (f) and averaged over 500 trajectories. The initial state is set to be close to a nematic phase ${\theta }_i={\sum }_{j=1}^i\pi [1+\mathrm{sgn}(J_j^{\mathrm{R}})]/2+{10}^{-3}{\eta }_i$, where ${\eta }_i=[0,2\pi )$ is a uniformly distributed random valuable.

Figure 15

Correlation function in the disordered state. (a) Spatial correlation function $C_x(x,t)$ and (b) temporal correlation function $C_t(t+t_w,t_w)$. We set $\gamma =1$ and the noise strength $\sigma =0.1$. We have averaged over 500 trajectories. Similarly to Fig. 14, we have set the initial state to be close to a nematic phase ${\theta }_i={\sum }_{j=1}^i\pi [1+\mathrm{sgn}(J_j^{\mathrm{R}})]/2+{10}^{-3}{\eta }_i$, where ${\eta }_i=[0,2\pi )$ is a uniformly distributed random valuable.

Figure 16

Time crystalline order-by-disorder phenomena with different intracommunity coupling strength. The intracommunity coupling strength is set to (a)–(c) $j_{\mathrm{AA}}=j_{\mathrm{BB}}=3$, (d)–(f) $j_{\mathrm{AA}}=5$, $j_{\mathrm{BB}}=2$, (g)–(i) $j_{\mathrm{AA}}=2$, $j_{\mathrm{BB}}=5$. (a),(d),(g) Angle difference $\mathrm{\Delta }\varphi$ dynamics, where a solid (thin) line represents the dynamics in the presence (absence) of noise. (b),(e),(h) Effective coupling $j_{ab}^{\star }(\varphi )$. (c),(f),(i) Phase ${\varphi }_a$ dynamics. While in (a)–(c), the chiral phase with $\mathrm{\Delta }{\varphi }_{*}=\pm \pi /2$ that satisfies $j_{\mathrm{AB}}^{\star }(\mathrm{\Delta }{\varphi }_{*})=-j_{\mathrm{BA}}^{\star }(\mathrm{\Delta }{\varphi }_{*})$ is selected, in (d)–(f) [(g)–(i)], as the effective coupling $j_{\mathrm{BA}}^{\star }(\mathrm{\Delta }\varphi )$ [$j_{\mathrm{AB}}^{\star }(\mathrm{\Delta }\varphi )$] is more strongly renormalized than $j_{\mathrm{AB}}^{\star }(\mathrm{\Delta }\varphi )$ [$j_{\mathrm{BA}}^{\star }(\mathrm{\Delta }\varphi )$], one always finds $j_{\mathrm{AB}}^{\star }(\mathrm{\Delta }\varphi )<-j_{\mathrm{BA}}^{\star }(\mathrm{\Delta }\varphi )$ [$j_{\mathrm{AB}}^{\star }(\mathrm{\Delta }\varphi )>-j_{\mathrm{BA}}^{\star }(\mathrm{\Delta }\varphi )$] that stabilizes the antialigned [aligned] phase characterized by the phase difference $\mathrm{\Delta }{\varphi }_{*}=\pi$ [$\mathrm{\Delta }{\varphi }_{*}=0$]. These results are all consistent with our analytical analysis [Eq. (b21)]. We set the noise strength $\sigma =1.5$, the number of spins $N_{\mathrm{A}}=N_{\mathrm{B}}=2000$, and the intercommunity coupling strength $j_{\mathrm{AB}}=-j_{\mathrm{BA}}=1$.