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Non-reciprocal phase transitions

Abstract

Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1,2,3,4,5,6, non-equilibrium systems7,8,9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13,14,15 and metamaterials16,17,18,19,20. Although wave propagation in non-reciprocal media has recently been closely studied1,16,17,18,19,20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.

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Fig. 1: Exceptional transitions: examples and mechanism.
Fig. 2: Phase diagrams and active time (quasi)crystals.
Fig. 3: Exceptional-point-enforced pattern formation and topological defects.
Fig. 4: A visual procedure to identify and analyse exceptional transitions.

Data availability

No data were generated during the course of this study.

Code availability

The computer code used in this study is available on Zenodo at https://doi.org/10.5281/zenodo.4605984 under the 2-clause BSD licence.

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Acknowledgements

We thank A. Alù, D. Bartolo, D. Christodoulides, A. Clerk, A. Edelman, A. Galda, M. Han, K. Husain, T. Kottos, Z. Lu, M. C. Marchetti, M.-A. Miri, B. Roussel, C. Scheibner, D. Schuster, J. Simon and B. van Zuiden. M.F. acknowledges support from a MRSEC-funded Kadanoff–Rice fellowship (DMR-2011854) and the Simons Foundation. R.H. was supported by a Grand-in-Aid for JSPS fellows (grant number 17J01238). V.V. was supported by the Complex Dynamics and Systems Program of the Army Research Office under grant number W911NF-19-1-0268 and the Simons Foundation. This work was partially supported by the University of Chicago Materials Research Science and Engineering Center, which is funded by National Science Foundation under award number DMR-2011854. This work was completed in part with resources provided by the University of Chicago’s Research Computing Center. Some of us benefited from participation in the KITP programme on Symmetry, Thermodynamics and Topology in Active Matter supported by grant number NSF PHY-1748958.

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M.F., R.H., P.B.L. and V.V. designed the research, performed the research, and wrote the paper.

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Correspondence to Vincenzo Vitelli.

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Extended data figures and tables

Extended Data Fig. 1 Codimensions of eigenvalue degeneracies.

This graph gives the codimension (codim) of twofold degeneracies of eigenvalues in different matrix spaces; see ref. 194. These degeneracies can be exceptional points (EP) or diabolic points (DP, also known as Dirac points). An identical graph can be drawn by replacing ‘real symmetric’ with ‘purely imaginary symmetric’, ‘Hermitian’ with ‘anti-Hermitian’ and ‘real’ with ‘imaginary’.

Extended Data Fig. 2 Many-body suppression of noise-activated chirality inversions.

a, A change in the sign of the angle Δϕ between the order parameters vA and vB (in blue and red) flips the chirality (clockwise or anticlockwise) of the chiral phase. Qualitatively, the two steady-state values ±Δϕc towards which the system relax correspond to the minima of an effective potential Uϕ), with a barrier ΔU separating these minima. The lifetime of the chiral phase is the average time τ separating two flips of chirality (represented in green), namely the Kramers escape time required to jump from one minimum to the other under the effect of noise. b, The standard deviations quantifying the fluctuations of the order parameters va in the chiral phase decrease approximately as \(1/\sqrt{N}\) with the number of agents N. The grey lines are equally spaced \(1/\sqrt{N}\) curves and are meant as a guide to the eye (not a fit). The data are obtained from simulations of the Kuramoto model equation (2) with JAA = JBB = 1,  JAB = 1,  JBA = −1.1, η = 8 × 10−2 and all-to-all couplings. The total duration is Tsimt = 4,000 with δt = 0.5, over which the standard deviation is computed.

Extended Data Fig. 3 Effect of non-conservative forces.

In this simplified pictorial representation, the order parameter (represented by a ball) evolves in a potential-energy landscape shaped like a sombrero. In a conservative system, the order parameter would relax straight to the bottom of the potential (dashed blue line). Here, transverse non-conservative forces push the order parameter in the direction defined by the bottom of the potential, leading to a curved trajectory (red continuous line) starting from the same initial condition. In the systems we considered, the non-conservative forces arise from the non-reciprocal coupling between two order parameters. This aspect is not captured by this simplified picture.

Extended Data Fig. 4 Phase diagram of the PT-symmetric non-reciprocal Kuramoto model and exceptional point in the spectrum of the Jacobian.

a, Phase diagram computed numerically from equation (15). The states are defined in Extended Data Table 1. b, The two most unstable eigenvalues λi = σi + iωi of L coalesce at  j+ ≈ 0.007. This value coincides with the transition from travelling waves (TW) to coherent states, marked by a red dashed line. Note that this coalescence occurs at λ = 0 (not at finite frequency nor at finite growth rate). The corresponding eigenvectors become collinear (this can be verified, for instance, by computing the determinant of the matrix of eigenvectors, that vanishes at the exceptional point). The imaginary parts ωi (not shown) are all zero. We have set  jAA = jBB = 1, ΔA = ΔB = 0.25 and ωA = ωB = 0. In b,  j = 0.1 (a similar behaviour is observed for neighbouring values of  j).

Extended Data Fig. 5 Hysteresis in the chiral Kuramoto model.

When chirality is explicitly broken, exceptional points have codimension two, that is, they are typically points in a two-dimensional parameter space. a, We plot the frequency Ω of the steady state of the Kuramoto model with explicitly broken PT symmetry as a function of the difference Δω = ωA − ωB between the two communities (also called detuning) and the deviation \({\rm{\delta }}{j}_{-}={j}_{-}-{j}_{-}^{{\rm{E}}{\rm{P}}}\) of the non-reciprocal part  j of the coupling between the communities from its value \({j}_{-}^{{\rm{EP}}}\) at the exceptional point. The system exhibits a region where two possible steady states with different properties coexist (the two steady states are the continuation of the clockwise and anticlockwise chiral phases present in the PT-symmetric case Δω = 0). This region (red triangle) starts at the exceptional point (red point) and its size increases with the amount of non-reciprocity (here \({j}_{-}^{{\rm{EP}}}\approx 0.2915 > 0\)). The system exhibits hysteresis in the coexistence region (red points). be, Slices from a at fixed δj (marked by green dotted lines in a). After the exceptional point, there is hysteresis/first-order (discontinuous) behaviour. In d, the hysteresis curve bends outwards near the transition. This is due to the oscillation of the norm of the order parameter (which we refer to as swap or periodic synchronization elsewhere) for large enough δj. This additional complication does not occur for moderate values of δj, such as in c. The solution of the dynamical system equation (15) were computed along lines at fixed δj, starting at large |δω| (in a region without phase coexistence) from a random initial condition. The solution (after convergence) was used as an initial value for the next point in the line with fixed δj. This procedure was carried out two times, starting from positive and negative large |δω|. We have set  j+ = 0.08,  jAA = jBB = 1, ΔA = ΔB = 0.25, ωA = ωB = Δω/2.

Extended Data Fig. 6 Non-reciprocal pattern formation.

We show a space-time density plot of the field u1(xt) in different phases, as well as snapshots of the fields u1(xt) and u2(xt) at time t = 200. a, We observe a disordered phase where both field vanish. b, An aligned phase where both patterns are static and in phase (superimposed). c, An antialigned phase where the patterns are static and completely out of phase. d, A chiral phase where the patterns move at constant velocity, either to the left or to the right (spontaneously breaking parity), and in which the fields have a finite phase difference, usually neither zero nor π. e, A swap phase where the patterns essentially jump by a phase π every period. f, A mix of the chiral and swap behaviours (as in the chiral phase (d), there is a spontaneously broken symmetry between left and right movers). The fields are obtained by direct numerical simulation of the coupled Swift–Hohenberg equations on a one-dimensional domain of size 2L with periodic boundary conditions, starting from random initial conditions. The simulations are performed using the open-source pseudospectral solver Dedalus219. We have used g = 0.25 in all cases. In a, r11 = r22 = −0.5 and r+ = r = 0.00. In the other cases, we have set r11 = r22 = 0.5 (bf) and r+ = 0.50, r = 0.00 (b); r+ = −0.50, r = 0.00 (c); r+ = 0.00, r = 0.25 (d); r+ = 0.87, r = 1.00 (e); r+ = 0.85, r = 1.00 (f).

Extended Data Fig. 7 Exceptional point in directional interface growth.

The spectrum of the Jacobian L corresponding to equation (20) exhibits an exceptional point at the transition between static patterns and travelling patterns with spontaneous parity breaking (that is, the patterns travel with equal probability to the left or to the right). The two most unstable eigenvalues λi = σi + iωi of L coalesce at μ1 ≈ 0.064 (red circle). This value coincides with the transition from a constant solution to travelling waves, marked by a red dashed line. The coalescence occurs at λ = 0 (not at finite frequency nor at finite growth rate), and the corresponding eigenvectors become collinear. Note that another exceptional point occurs near μ1 ≈ 0.014 (green circle), but with a strictly negative growth rate: this does not correspond to a bifurcation. We also show the dephasing Δϕ = 2ϕ1 − ϕ2 between the amplitudes, which undergoes a pitchfork bifurcation; the direction of motion of the pattern is set by Δϕ. We have set α = β = γ = δ = 1, ε = +1 and μ2 = −0.1.

Extended Data Table 1 An O(2) ‘Rosetta stone’

Supplementary information

Supplementary Information

This file contains derivations of the results presented in the main text and additional discussions.

Supplementary Video 1

Demonstration with programmable robots. See Sec. XIV of the Supplementary Notes.

Supplementary Video 2

Evolution of the order parameters vA (blue) and vB (red) in the time-dependent phases (chiral, swap, chiral+swap) computed from the dynamical system Eq. (S84).

Supplementary Video 3

Molecular dynamics simulation of the microscopic non-reciprocal flocking model. See Sec. V B of the Supplementary Notes. In the bottom part, we show the instantaneous order parameter obtained by averaging the individual directions of the self-propelled particles.

Supplementary Video 4

Numerical simulation of the hydrodynamic field theory showing pattern formation at fixed density when the incompressibility constraint is not enforced. See Sec. XV of the Supplementary Notes.

Supplementary Video 5

Numerical simulation of the hydrodynamic field theory showing pattern formation with the incom- pressibility constraint enforced. See Sec. XV B of the Supplementary Notes.

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Fruchart, M., Hanai, R., Littlewood, P.B. et al. Non-reciprocal phase transitions. Nature 592, 363–369 (2021). https://doi.org/10.1038/s41586-021-03375-9

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