Field theory for the dynamics of the open $O\left(N\right)$ model

A field theory approach for the nonequilibrium relaxation dynamics in open systems at late times is developed. In the absence of conservation laws, all excitations are subject to dissipation. Nevertheless, ordered stationary states satisfy Goldstone's theorem. It implies a vanishing damping rate at small momenta, which in turn allows for competition between environment-induced dissipation and thermalization due to collisions. We derive the dynamic theory in the symmetry-broken phase of an $O\left(N\right)$-symmetric field theory based on an expansion of the two-particle irreducible (2PI) effective action to next-to-leading order in $1/N$ and highlight the analogies and differences to the corresponding theory for closed systems. A central result of this approach is the systematic derivation of an open-system Boltzmann equation, which takes a very different form from its closed-system counterpart due to the absence of well-defined quasiparticles. As a consequence of the general structure of its derivation, it applies to open, gapless field theories that satisfy certain testable conditions, which we identify here. Specifically for the $O\left(N\right)$ model, we use scaling analysis and numerical simulations to show that interactions are screened efficiently at small momenta and, therefore, the late-time evolution is effectively collisionless. This implies that fluctuations induced by a quench dissipate into the environment before they thermalize. Goldstone's theorem also constrains the dynamics far from equilibrium, which is used to show that the order parameter equilibrates more quickly for quenches preserving the $O\left(N\right)$ symmetry than those breaking it explicitly.

Figure 1

Decay rate and excitation energy of the Goldstone mode of the open $O\left(N\right)$ model. At short times excitations at high momenta are well described by the closed system. However, the finite decay rate at high momenta at times $t\gg 1/\gamma$ only leaves long-lived, overdamped excitations at small momenta. The resulting slow dynamics is qualitatively different from that of the closed system, as is evident from the change of the dynamical exponent from $z=1$ to $z=2$.

Figure 2

The theory of open system dynamics is developed along the same lines as that for closed systems. While the semiclassical limit of high occupations is the same for both cases, gapless dissipative modes in open systems require a separate treatment from closed systems for all further steps towards a simple, physically insightful theory of the time evolution, see Sec. 3. This schematic can be taken as a guide to this paper: Sec. 4 deals with the semiclassical action. Sections 5a and 5b explain the steps to the memoryless equations of motion. Although numerically validated in some cases, there is no simple condition for this step in closed systems. A projection to an open-system Boltzmann equation is discussed in Sec. 8.

Figure 3

At each order in $1/N$ 2PI requires far fewer diagrams than 1PI. This becomes apparent already at leading order, where due to the restriction to skeleton diagrams (i.e., those without self-energy insertions) only a single diagram contributes to the effective 2PI action. Bold lines indicate dressed Green's functions and the thin dashed line represents the contact interaction $\sim \frac{\lambda }{3N}\delta (x-x^{\prime })$.

Figure 4

The diagrammatic representation of ${\mathrm{\Gamma }}_2$ up to NLO($1/N$) is the sum of all interaction rings with any number of screening bubbles inserted. In each diagram, one bubble may be opened with one Green's function being replaced by two fields if the cut diagram involves at least two loops. Here we show the first three terms of this series.

Figure 5

Coupled Dyson equations in 2PI at next-to-leading order in $1/N$ with symmetry factors stated explicitly. To write the self-energies in a compact form we distinguish between two types of dressed interactions indicated as wavy and zigzag lines. The last line corresponds to the equation of motion of the field expectation value $\varphi \equiv {\varphi }_{\text{cl}}$.

Figure 6

Contribution missing from the transversal self-energy responsible for the leading contribution to the Goldstone mass of order $\mathcal{O}\left(N^{-2}\right)$.

Figure 7

Feynman diagrams that need to be added to the effective action to suppress the Goldstone mass in 2PI to $\mathcal{O}\left(N^{-3}\right)$.

Figure 8

Comparison between the two limiting cases of small and large momenta. At small momenta (a), the correlations of ${\mathrm{\Sigma }}^K(t_1,t_2,\mathbf{k})$ are very short compared to those in the spectral function $\mathcal{A}=G^A-G^R$. Hence the dependence of ${\mathrm{\Sigma }}^K(t_1,t_2,\mathbf{k})$ on $t_1-t_2$ can be approximated as $\delta (t_1-t_2)$, which in frequency space translates to ${\mathrm{\Sigma }}^K(t,\omega ,\mathit{k})={\mathrm{\Sigma }}^K(t,\mathbf{k})$. However, its algebraic dependence on $t$ remains important as the narrow spectrum implies a long memory. In the opposite case (b) excitations are very short-lived, therefore the slow variation of ${\mathrm{\Sigma }}^K(t,\omega ,\mathit{k})$ in time can be neglected, but its relatively long correlations in $t_1-t_2$ render its frequency dependence in Wigner coordinates important. The two cases therefore can be thought of as a piano with a pressed damper pedal (a) or una corda pedal (b).

Figure 9

Comparison between the full integral $I(t,\omega ,\mathit{k})$ for $t=100, \alpha =-3/2$, and $z=2$ with the zeroth-order Wigner approximation $I_0(t,\omega ,\mathit{k})$. In (a), we chose the momentum $\left|\mathbf{k}\right|=\sqrt{t}$, where the Wigner expansion is locally not yet a good approximation as it misses the fast oscillations $\sim cos\left(2\omega t\right)$. At the same time at the larger momentum $\left|\mathbf{k}\right|=\sqrt{10t}$ shown in (b) the system has already forgotten about the quench and is therefore well approximated by the zeroth-order Wigner expansion at all frequencies. At small momenta, one thus has to average over a frequency range $\gg 1/t$ for the Wigner expansion to become a good approximation.

Figure 10

Comparison between the time evolution following a weak temperature quench in the collisionless approximation (leading order in $1/N$) and the dynamics including scattering in the memoryless approximation of Eq. (39). Due to the strong screening of interactions at long wavelengths, the asymptotic behavior of the collisionless description in the overdamped approximation (light green) agrees extremely well with the expansion to next-to-leading order in $1/N$ (dark red). Both differ at early times from the collisionless evolution without overdamped approximation (blue), see insets. Both the order parameter (top) and the mass of the transversal mode (middle) follow the prediction Eq. (58). The bottom panel shows the evolution of the transversal momentum distribution. While collisions increase the density at high momenta (full lines) compared to the collisionless (dashed) result, the decay of the thermal peak as well as the momentum scale with the largest number of excitations $k_{\text{Coll}}$ (see inset) agrees well between the two methods. The inverse screening length $1/{\xi }_{\parallel }$ is shown as gray vertical line and we used the parameters $\lambda =T=\gamma =D=1, N=10$, and ${\varphi }_{\text{th}}=\sqrt{5}$.

Figure 11

The leading order of ${\mathrm{\Gamma }}_1$ in the expansion in $1/N$ consists of all tadpole diagrams with at least two loops. Note, that the bare Green's function (thin line) already contains the coupling to the field $\varphi$ [see Eq. (A2)]. In addition, the dressed Green's functions (thick lines) contain self-energy corrections in the form of tadpole diagrams. The thin dashed line indicates the bare vertex. Symmetry factors and Keldysh structure have been suppressed.

Figure 12

The diagrammatic representation of ${\mathrm{\Gamma }}_1$ up to NLO($1/N$) is the sum of all interaction rings with any number of screening bubbles inserted. Note, that each bubble may be opened and replaced by fields as long as the final diagram contains at least two loops. The corresponding one-loop diagrams are already included in ${\mathrm{Tr}}_{\mathcal{C}}ln{G}^{-1}$. Here we show the first three terms of this series.

Figure 13

Diagrammatic representation of the 1PI transversal Green's function $G_{\perp }$ in next-to-leading order in $1/N$. For notational simplicity, we have suppressed the Keldysh structure. One may check that the self-energy is the sum of all one-particle irreducible diagrams that are no smaller than $1/N$. In particular, we point out that the restriction to irreducible self-energies implies a difference between the dressed Hartree interaction $\mathrm{\Pi }$ and the dressed Fock vertex $W$. By making all interactions explicit to ease the identification of diagrams in Eq. (A9) we break with our previous convention as here the bare Green's function lacks interactions with the field (last diagram in the first line).

Figure 14

Time evolution of the transverse mode following the weak temperature quench discussed in the main text. The comparison shows the collisionless approximation (leading order in $1/N$) (light green), the collisionless evolution without overdamped approximation (blue), and the scattering dynamics in the memoryless approximation of Eq. (39) (dark red). At momenta larger than the inverse screening length, $k>1/x{i}_{\parallel }$ (top), all methods agree at late times. At smaller momenta $k\ll 1/{\xi }_{\parallel }$, however, the effect of collisions dominates at late times (bottom). The same parameters as in the main text are used $\lambda =T=\gamma =D=1, N=10$, and ${\varphi }_{\text{th}}=\sqrt{5}$ corresponding to an inverse screening length $1/\xi \approx 0.4$.

Figure 15

Momentum dependence of the excess density created by the weak temperature quench (63). Analogously to Fig. 10, the dashed lines correspond to the collisionless approximation and solid lines to the calculation at next-to-leading order in $1/N$. The same parameters as in the main text are used $\lambda =T=\gamma =D=1, N=10$, and ${\varphi }_{\text{th}}=\sqrt{5}$ corresponding to an inverse screening length $1/\xi \approx 0.4$, which is indicated as a gray vertical line.