Nonequilibrium universality of the nonreciprocally coupled $\mathbf{O(n_1) \times O(n_2)}$ model

Abstract

In this work, we investigate an important class of nonequilibrium dynamics in the form of nonreciprocal interactions. In particular, we study how nonreciprocal coupling between two $O(n_i)$ order parameters (with $i=1,2$) affects the universality at a multicritical point, extending the analysis of [J.T. Young et al., Phys. Rev. X 10, 011039 (2020)], which considered the case $n_1 = n_2 = 1$, i.e., a $\mathbb{Z}_2 \times \mathbb{Z}_2$ model. We show that nonequilibrium fixed points (NEFPs) emerge for a broad range of $n_1,n_2$ and exhibit intrinsically nonequilibrium critical phenomena, namely a violation of fluctuation-dissipation relations at all scales and underdamped oscillations near criticality in contrast to the overdamped relaxational dynamics of the corresponding equilibrium models. Furthermore, the NEFPs exhibit an emergent discrete scale invariance in certain physically-relevant regimes of $n_1,n_2$, but not others, depending on whether the critical exponent $ν$ is real or complex. The boundary between these two regions is described by an exceptional point in the renormalization group (RG) flow, leading to distinctive features in correlation functions and the phase diagram. Another contrast with the previous work is the number and stability of the NEFPs as well as the underlying topology of the RG flow. Finally, we investigate an extreme form of nonreciprocity where one order parameter is independent of the other order parameter but not vice versa. Unlike the $\mathbb{Z}_2 \times \mathbb{Z}_2$ model, which becomes non-perturbative in this case, we identify a distinct nonequilibrium universality class whose dependent field similarly violates fluctuation-dissipation relations but does not exhibit discrete scale invariance or underdamped oscillations near criticality.

Figures (7)

• Figure 1: Interaction vertices. Thin black (thick cyan) lines correspond to the first (second) field and solid (dashed) lines correspond to the classical (response) field. The inclusion of the circles is to illustrate the pairing of legs, i.e., the two legs without circles are involved in one dot product while the two legs with circles are involved in the other. For each vertex, we must sum over all of the components contributing to the dot product for each leg pair. The vertices correspond to (a) $u_1 |\boldsymbol{\Phi}_1|^2 \boldsymbol{\Phi}_1 \cdot \tilde{\boldsymbol{\Phi}}_1$, (b) $u_2 |\boldsymbol{\Phi}_2|^2 \boldsymbol{\Phi}_2 \cdot \tilde{\boldsymbol{\Phi}}_2$, (c) $u_{12} |\boldsymbol{\Phi}_2|^2 \boldsymbol{\Phi}_1 \cdot \tilde{\boldsymbol{\Phi}}_1$, and (d) $\sigma u_{12} |\boldsymbol{\Phi}_1|^2 \boldsymbol{\Phi}_2 \cdot \tilde{\boldsymbol{\Phi}}_2$.
• Figure 2: One-loop corrections to (a,b) $r_1 \boldsymbol{\Phi}_1 \cdot \tilde{\boldsymbol{\Phi}}_1$, (c,d) $u_1 |\boldsymbol{\Phi}_1|^2 \boldsymbol{\Phi}_1 \cdot \tilde{\boldsymbol{\Phi}}_1$, and (e-h) $u_{12} |\boldsymbol{\Phi}_2|^2 \boldsymbol{\Phi}_1 \cdot \tilde{\boldsymbol{\Phi}}_1$. Analogous diagrams for $r_2 \boldsymbol{\Phi}_2 \cdot \tilde{\boldsymbol{\Phi}}_2$, $u_2 |\boldsymbol{\Phi}_2|^2 \boldsymbol{\Phi}_2 \cdot \tilde{\boldsymbol{\Phi}}_2$ and $\sigma u_{12} |\boldsymbol{\Phi}_1|^2 \boldsymbol{\Phi}_2 \cdot \tilde{\boldsymbol{\Phi}}_2$ can be obtained by switching thin black and thick cyan lines. Two-loop corrections to (i-k) $\zeta_1$ and $D_1$ as well as (l,m) $\zeta_1 T_1$. Analogous diagrams for $\zeta_2,D_2$, and $\zeta_2 T_2$ can be obtained by switching thin black and thick cyan lines. In these diagrams, the circles indicate propagators corresponding to the same component of a given field involved in a dot product and which must be summed over.
• Figure 3: Fixed point behavior as a function of $n_1, n_2$ with qualitative stability to $O(\epsilon)$ aside from $\tilde{u}_{i_R}^* < 0$ fixed points. The top plots indicate different regions of fixed point and stability behavior, separated by three types of boundaries, labeled I, II, and III, indicated by dashed, solid, and dotted lines, respectively. Shading indicates which region of $n_1, n_2$ we consider. The corresponding bottom plots illustrate qualitatively the RG flow diagrams in the $g_{12}$-$g_{21}$ plane, although the full flow occurs in a five-dimensional space. $\mathcal{N}$ denotes the NEFPs, $\mathcal{H}$ the Heisenberg fixed points with $O(n_1+n_2)$ symmetry, $\mathcal{B}$ the biconical fixed points, and $\mathcal{D}$ the decoupled fixed points. The dotted green lines in the bottom plots indicate parameters corresponding to effective equilibrium behavior, so $\mathcal{H},\mathcal{B}, \mathcal{D}$ lie along these lines. Since stability is only known to first-order in $\epsilon$ along directions which preserve $g_{21}/g_{21}$, we use filled black arrows to indicate known stability at this order and gray arrows to indicate the anticipated stability at higher orders. Given the fact that the system cannot flow through $g_{12} = 0$ or $g_{21} = 0$, we expect each quadrant (minus the axes) to broadly define the region of attraction for the corresponding stable fixed point, provided one exists and the system remains in the perturbative regime. (a) In this region, both NEFPs are present and stable at $\mathcal{O}(\epsilon)$. Whether $\mathcal{B}$ or $\mathcal{H}$ is stable is determined by the values of $n_1, n_2$. (b) In this region, there are no physically valid NEFPs, and criticality is described by $\mathcal{D}$. (c) In this region, one of the two NEFPs has diverged, leaving only the other NEFP, which is stable at $\mathcal{O}(\epsilon)$. Whether $\mathcal{B}$ or $\mathcal{H}$ is stable is determined by the values of $n_1, n_2$. (d) In this region, both NEFPs are in the same quadrant, with one stable and one unstable at $\mathcal{O}(\epsilon)$. (e) In this region, one of the two NEFPs has diverged, leaving only the other NEFP, which is unstable at $\mathcal{O}(\epsilon)$.
• Figure 4: Behavior of right (solid) and left (dashed) eigenvectors of the flow defined by $\mathcal{R}$ for each of the three possible cases at a NEFP. The top plots illustrate the behavior of the magnitude of the $r_i$ components while the bottom plots illustrate the behavior of the relative complex phase $\vartheta$ of the $r_i$ components. (a) Real-valued $\nu$: two different real eigenvectors. The analogous plots for equilibrium fixed points can be obtained by applying a reflection to the eigenvectors (both left and right) associated with one eigenvalue, corresponding to a shift of $\pi$ in $\vartheta$. (b) Exceptional point which occurs between real- and complex-valued $\nu$ regions: both eigenvectors coalesce into a single eigenvector. (c) Complex-valued $\nu$: two complex eigenvectors which are conjugate to one another.
• Figure 5: Phase diagrams associated with the NEFPs. The white region indicates the disordered phase with $\langle \boldsymbol{\Phi}_1 \rangle = \langle \boldsymbol{\Phi}_2\rangle = 0$, the red vertically shaded region (blue horizontally shaded region) one of the singly-ordered phases with $\langle \boldsymbol{\Phi}_1 \rangle \neq 0$ ($\langle \boldsymbol{\Phi}_2 \rangle \neq 0$), and the purple square shaded region the doubly-ordered phase with both $\langle \boldsymbol{\Phi}_1 \rangle, \langle \boldsymbol{\Phi}_2 \rangle \neq 0$. The behavior of the phase diagram depends on the exponent $\nu$. (a) When $\nu$ is real, the phase diagram generally behaves like for the equilibrium coupled fixed points, where the phase boundaries approach the multicritical point tangentially to $\mathbf{v}_{a}$, the right eigenvector of $\mathcal{R}$ associated with the correlation length exponent $\nu$, like a polynomial. (b) In the transition between these two scenarios as a function of $n_1$, $n_2$, the flow of $r$ undergoes an exceptional point. The corresponding phase boundaries approach the multicritical point like $r \log r$. Note that the phase boundaries approach from the same side of $\mathbf{v}_e$, the only right eigenvector of $\mathcal{R}$, a consequence of the coalescence of the eigenvectors. (c) When $\nu$ is complex, the eigenvectors of $\mathcal{R}$ are as well. As a result, the phase diagram exhibits logarithmic spirals with discrete scale invariance. In general, these spirals are skewed along a vector $\tilde{\mathbf{v}}_2$, which is defined by a matrix $\tilde{\mathcal{M}}$ [cf. Eq. (\ref{['eq:tildeS']})] that transforms the skewed spirals into isotropic spirals via a basis change.